Degrees of Essentiality for Secants of Knots

DEGREES OF ESSENTIALITY FOR SECANTS OF KNOTS

vorgelegt von Dott.ssa Magistrale in Matematica Silvia De Toﬀoli aus Venedig

Von der Fakultät II - Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr.rer.nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Fredi Tröltzsch Berichter/Gutachter: Prof. John M. Sullivan, Ph.D. Berichter/Gutachter: Prof. Paolo Bellingeri, Ph.D.

Tag der wissenschaftlichen Aussprache: 04/06/2013

Berlin 2013 D 83

Degrees of Essentiality for Secants of Knots

by Silvia De Toﬀoli

Figure 1: A knot and its reﬂection

To (SD)2

Abstract

Knots are simple closed curves in space. Secants of knots are straight segments intersecting a knot in two points. First, we study essential and strongly essential secants. Essential secants were ﬁrst introduced by Kuperberg and strongly essential secants by Denne. We introduce another degree of essentiality for secants, that is between the mentioned ones: non-peripherality. Then, we introduce a new and stronger concept: n-essentiality. To deﬁne it, we use the theory of n-fold cyclic branched covers of the 3-sphere over a knot, both geometrically and algebraically. We prove that for all n ∈ N a n-essential secant is strongly essential and moreover it is kn-essential for all k ∈ N. We also turn our focus to the diagrammatic level, considering the essentiality of the secants corresponding to the crossings in a knot diagram. This is a new perspective that leads to a combinatorial treatment of secants. Given a knot diagram, to every crossing corresponds a secant, the one perpendicular to the plane of projection connecting the two strands that intersect at the crossing. Our main result is that in a minimal diagram of a alternating knot all crossings are strongly essential. Other results are that in any diagram of a non-trivial knot there must be at least three 2-essential crossings and that all crossings in a minimal diagram of a rational knot are 2-essential.

Zusammenfassung

Verschiedene Grade von Essenzialität für Sekanten von Knoten

Knoten sind geschlossene, einfache Kurven im Raum. Sekanten sind gerade Strecken, die zwei Punkte auf einem Knoten verbinden. Wir untersuchen essenzielle und stark essenzielle Sekanten. Essenzielle Sekanten wurden zuerst von Kuperberg und stark essenzielle Sekanten von Denne eingeführt. Wir führen einen anderen Grad von Essenzialität ein, der zwischen den beiden oben genannten liegt: Nichtperipheralität. Danach führen wir einen neuen und stärkeren Begriﬀ ein: n-Essenzialität. Um ihn zu deﬁnieren benutzen wir die Theorie der n-fachen, zyklischen, über dem Knoten verzweig- ten Überlagerungen der 3-Sphäre, sowohl geometrisch als auch algebraisch. Wir beweisen, dass jede n-essenzielle Sekante stark essenziell ist, und jede n-essenzielle Sekante auch kn-essenziell ist. Wir richten unsere Aufmerksamkeit auch auf die diagrammatische Ebene und be- trachten die Essenzialität von Sekanten, die zu Kreuzungen in einem Knotendiagramm gehören. Das ist eine neue Sichtweise, die zu einer kombinatorischen Behandlung von Sekanten führt. In einem Knotendiagramm entspricht jede Kreuzung einer Sekante, und zwar der zur Projektionsebene senkrechten Sekante, die die sich kreuzenden Stränge ver- bindet. Unser Hauptresultat besagt, dass in einem minimalen Diagramm eines alternieren- den Knotens alle Kreuzungen stark essenziell sind. Außerdem zeigen wir, dass in jedem Diagramm eines nicht-essenziellen Knotens mindestens drei 2-essenzielle Kreuzungen exi- stieren müssen und dass alle Kreuzungen eines minimalen Diagrammes eines rationalen Knotens 2-essenziell sind.

Contents

Abstract vii

Zusammenfassung ix

List of Figures xiii

Notation and conventions xvii

0 Introduction1 0.1 General Introduction and Motivations...... 1 0.2 Outline...... 2 0.3 Acknowledgements...... 6

1 Mathematical Knots9 1.1 Knots and Knot Types...... 9 1.2 Knot Diagrams...... 14 1.3 Knot Invariants...... 20

2 Essential and Strongly Essential Secants 29 2.1 Deﬁnitions...... 29 2.2 Essential Crossings...... 43 2.3 Secants Under Knot Composition...... 47 2.4 Smoothings of a Crossing...... 54

xi xii Contents

3 Secants of Alternating Knots 59 3.1 Hyperbolic Knots...... 60 3.2 (2, q)-Torus Knots...... 79 3.3 Composite Knots...... 82

4 n-Essential Secants 85 4.1 Cyclic Branched Covers of Knots...... 85 4.2 n-Essentiality...... 92 4.3 The Double Branched Cover and its Fundamental Group...... 99 4.4 2-Essentiality, a Combinatorical Approach...... 107

5 2-essentiality and Families of Knots 117 5.1 Rational Knots...... 117 5.2 The Torus Knots T (2, q), T (3, q) and T (5, q) ...... 126

Appendix A: Topology Miscellanea 131

Bibliography 133 List of Figures

1 A knot and its reﬂection...... iii

1.1 Knots and links...... 10 1.2 Pulling a knot tight...... 12 1.3 A polygonal knot...... 13 1.4 A wild knot...... 13 1.5 Two unknots...... 14 1.6 A bad illustration of the unknot...... 15 1.7 A messy projection...... 16 1.8 Three equivalent diagrams...... 17 1.9 Another equivalent diagram...... 17 1.10 Two non-trivial diagrams of the trivial knot...... 18 1.11 The three Reidemeister moves...... 18 1.12 Nugatory Crossings...... 19 1.13 A positive crossing (left) and a negative crossing (right)...... 20

1.14 The knot 817 ...... 20 1.15 An alternating diagram (left) and a non-alternating diagram (right).... 21

1.16 The knot 819 ...... 21 1.17 Crossing change...... 22 3 1.18 The map f : X → S ...... 24 1.19 A composite knot diagram...... 25 1.20 The knot T (3, 5) ...... 26 1.21 Satellite knots...... 27

xiii xiv List of Figures

2.1 A secant arc corresponding to the secant S ...... 30 2.2 An inessential secant (a) and an essential secant (b)...... 33

2.3 The loop p1 ...... 34 2.4 An essential but peripheral arc...... 35 2.5 A simply inessential secants...... 35 2.6 An insessential but not simply inessential secant...... 36 2.7 A loop ‘around’ a secant-arc...... 38 2.8 Strong essentiality...... 39

2.9 Notation for the generators of π1(X) ...... 40

2.10 Generators of π1(X) ...... 41 2.11 Sliding a loop through a crossing...... 41 2.12 Region relation...... 43 2.13 A positive and a negative crossing...... 44 2.14 The loops around a positive crossing (a) and around a negative crossing (b) 45 2.15 A knot exterior as a ball minus a tube...... 48 2.16 The neighborhood of the boundary to be glued...... 49 2.17 A secant in a composite knot...... 51 2.18 “Disks" constructions...... 52 2.19 Swallow-follow torus...... 53 2.20 Desingularization of a crossing...... 54 2.21 A crossing which give rise to a split link...... 55 2.22 Local diﬀerence between D and D0 ...... 56 2.23 A twist knot...... 58

3.1 A ﬂype...... 60 2 3.2 Geodesics in the disk model of H ...... 61 2 3.3 Geodesics in the upper half plane model of H ...... 62 3.4 Loxodromic and elliptic isometries on the disk model...... 63 3.5 A parabolic isometry on the upper half space model...... 63 3.6 The complement (a) and the exterior (b) of a hyperbolic knot...... 66 3.7 Horoballs in the disk model (a) and half plane model (b)...... 67 List of Figures xv

3.8 A knot diagram with a checkerboard coloring...... 72 3.9 A path in X (a) and one of its lifts (b)...... 75 3.10 A peripheral path in X (a) and one of its lifts (b)...... 75 3.11 An arc homotopic to a geodesic connecting the centers of two horoballs in 3 H ...... 77 3.12 Standard diagram of a (2, q)-torus knot...... 79 3.13 Flypes in a braid diagram of a (2, q)-torus knot...... 80 3.14 Flypes in a braid diagram of a (2, q)-torus knot...... 80 3.15 Flypes in a braid diagram of a (2, q)-torus knot...... 81

3.16 Wirtinger presentation of π1(X) ...... 82 3.17 A loop relative to a crossing...... 82

2 4.1 D as a branched cover of itself over a point...... 88 2 4.2 S as a branched cover of itself over two points...... 89 4.3 Minimal diagrams of the trefoil knot...... 95 4.4 Loops corresponding to secants of the trefoil...... 96 2 −1 −2 3 −1 −3 4.5 The braids σ1σ2σ1 σ2 and σ1σ2σ1 σ2 ...... 97 −1 2 −1 4.6 The braid σ2 σ1σ2 ...... 97 10 −1 −10 −1 10 −9 4.7 The braids σ1σ2 σ1 σ2 and σ2 σ1 σ2 ...... 98 4.8 A positive and a negative crossing...... 101 4.9 The loops around a positive and a negative crossing...... 102

4.10 Equivalent loops in Σ2(K) for a positive (left) and a negative crossing (right)104 4.11 A knot and its associated graph...... 104 4.12 A band in a knot diagram...... 105 4.13 Loops in a band...... 106 4.14 A ﬂype...... 107 4.15 A tree of regions ending in nugutory crossings...... 108 4.16 Two opposite regions are equal...... 109 4.17 The regions around a crossing...... 109 4.18 A composite diagram...... 110 4.19 The edge AB ...... 111 xvi List of Figures

4.20 Two vertices share three regions...... 111 4.21 Two crossings sharing all the 4 regions...... 112 4.22 Diagram of the trefoil knot with three 2-essential crossings and one inessential crossing...... 114 4.23 Graphs corresponding to prime knot diagrams...... 114 4.24 Regions in a graph corresponding to a five crossing prime knot diagram. 115 4.25 Diagrams of the trefoil with three, four and five 2-essential crossings... 116 4.26 Graph corresponding to a non-trivial knot that might only have five 2- essential crossings...... 116

5.1 The diagram C(a1, . . . , an) ...... 118

5.2 The diagram C(a1, . . . , an) ...... 121

5.3 Loops around crossings in the diagram C(a1, . . . , an) ...... 122 5.4 Region types...... 123 5.5 The diagram C(4, −1, 1, 6) ...... 125 5.6 Braid diagram of the torus knot T (3, 7) ...... 126 5.7 Equivalent loops around crossings in the diagram of T (3, 7) ...... 127 5.8 Braid diagram of the torus knot T (5, 6) ...... 128 5.9 Equivalent loops around crossings in the braid diagram of T (5, 6) ..... 128 5.10 Equivalent crossings in the braid diagram of T (5, 6) ...... 128 5.11 Equivalent crossings in the braid diagram of T (5, 6) ...... 129 5.12 Regions in the diagram of T (5, 6) ...... 129 Notation and conventions

n 3 3 S : the n-dimensional sphere. S ' R ∪ ∞ is oriented in the canonical way (coherently 3 with the orientation of R given by the right-hand rule).

n n−1 D : the n-dimensional disk, whose boundary is S .

1 I: the unit interval [0, 1], which is homeomorphic to D .

2 1 1 T : the torus, a surface homeomorphic to S × S .

2 1 T : the solid torus, a 3-dimensional manifold homeomorphic to D ×S , whose boundary is a torus.

∂M: the boundary of a smooth manifold M. If the space M is oriented, then also its boundary will be oriented using the following rule outside normal ﬁrst.

◦ M: the interior of the manifold M.

Hn(X): the n-th singular homology group with integer coeﬃcients.

π1(X, x0): the fundamental group of a topological space X with respect to the base-

point x0.

π1(X): the fundamental group of a path-connected topological space X with respect to an arbitrary base-point. (It is well deﬁned up to group isomorphism: For all

x0, x1 ∈ X, π1(X, x0) ' π1(X, x1).)

[y]: the homotopy class of y ; [y] ∈ π1(X).

3 3 3 3 K: a smooth oriented knot in R . We consider R to be embedded in S ' R ∪ {∞}.

xvii xviii List of Figures

D: a knot diagram.

N(K): a closed tubular neighborhood of K, which is therefore homeomorphic to a solid torus.

◦ 3 3 X: the exterior space of a knot K in S , X ' S \ N(K).

3 3 C: the complement of a knot K in S , C ' S \ K.

G: the fundamental group of the exterior X of a knot K.

X˜n: the n-fold cyclic cover of X.

3 Σn(K): the n-fold cyclic cover of S branched along the knot K.

S: a secant of a knot K.

A: a secant arc in a knot exterior X.

lA: the loop in X around A with linking number 0 with the knot.

I: the set of all strands of knot diagram D.

c = (a, b1, b2), where a, b1, b2 ∈ I: a crossing of D.

lc: the loop in X around the crossing c. Chapter 0

Introduction

0.1 General Introduction and Motivations

Knot theory is a mathematical domain that originated in the late nineteenth century and which has seen great development in recent years. It is a branch of topology that focuses on mathematical knots, which are embeddings of circles in a 3-dimensional space. We can think of mathematical knots as abstractions of physical knots, such as the knots we use to tie our shoes: However, unlike physical knots, these have no thickness (i.e. they are one-dimensional, like circles), they are closed (the ends of the strand are glued together) and they are embedded in a 3-dimensional mathematical space. The ﬁrst interest in knot theory was one of physical nature. For example, Lord Kelvin conceived atoms as knotted vortices in the ether. His idea was that through knots we could gain information about the nature of diﬀerent atoms. When this theory was proven wrong, knots were neglected until recent developments in topology showed their mathematical importance. Nowadays, knots are of great importance in topology for many reasons; for example, we can use knots and rational numbers as codes to create all 3-dimensional manifolds from the 3-sphere (using a process known as Dehn surgery). Moreover, natural science has discovered a new interest in knots. To give an example, biologists rely on the mathematical theories of certain classes of knots in order to understand the behaviour of knotted DNA. Perhaps the greatest problem in knot theory is in recognizing when two knots are

1 2 Introduction equivalent, that is when one can be deformed into the other. In fact in knot theory we are not generally interested in the particular geometric shape of a knot but rather on its “knottedness”, that is, in the way a knot is knotted. In order to work with knots, we associate to them knot diagrams, which are regular projections of knots on a surface with local height information at their intersection points. One approach to the problem of distinguishing knot types is to associate mathematical structure to the corresponding knot diagrams. In our speciﬁc research, we focus on secants of knots, which can be associated either to a knot in space or to a diagram in a surface. A secant of a knot is a straight segment which joins two points of the knot. First we characterize the already introduced concepts of essential and strongly essential secants and then we introduce new grades of essentiality. The goal of this thesis is to understand the behavior of secants with various degrees of essentiality. Its results can be used to derive various properties of knots. This research is also relevant in geometric knot theory to ﬁnd various properties of geometric knots, for example lower bounds for ropelength and distortion (some results have been already achieved, see [20] and [21]). We also turn our focus to the diagrammatic level, considering the essentiality of secants and arcs corresponding to a crossing in a knot diagram. This is a new perspective that leads to a combinatorial treatment of secants. Given a knot diagram, to every crossing corresponds a secant, the one perpendicular to the plane of projection connecting the two strands that intersect at the crossing. Our main result is that in a minimal diagram of a alternating knot all crossings are strongly essential. Other results are that in any diagram of a non-trivial knot there must be at least three 2-essential crossings and that all crossings in a minimal diagram of a rational knot are 2-essential.

0.2 Outline

• In Chapter1 we deﬁne mathematical knots and state some basic well-known properties of them. All the material in this chapter is a review of already established facts that we will use in the following discussion.

• In Chapter2 we introduce secants and begin classifying them. Outline 3

In Section 2.1 we define essential and strongly essential secants. Equivalent definition have been already introduced, the one of essential secant by Kuperberg [32] and the one of strongly essential secant by Denne [19] (and then developed by Denne, Diao and Sullivan [20]). We approach the problems related to secants of knots by working not only with secants, but also with secant arcs, which are proper arcs in the knot exterior. Then, we introduce another degree of essentiality that is between the previous ones: non-peripherality (Definition 2.5). In this section we also introduces the well known Wirtinger presentation of the knot group. We prove a lemma (Lemma 2.25) using this presentation that will be useful in analyzing essential secants.

Then, in Section 2.2 we turn our focus to the diagrammatic level considering the essentiality of secants and arcs corresponding to a crossing in a knot diagram. This is a new perspective that leads to a more combinatorial treatment of secants. Given a knot diagram, to every crossing corresponds a secant: the one perpendicular to the plane of projection connecting the two strands that intersect at the crossing. We say that a crossing is essential if the corresponding secant is essential. This notion allows us to work directly with knot diagrams. We prove that any diagram of a non-trivial knot must have at least one strongly essential crossing (Theorem 2.31).

In Section 2.3 we begin by stating well-established facts about knot composition, and then we apply them in order to study the behaviour of strongly essential and essential crossings under knot composition. We ﬁrst present the standard calculation of the knot group of a composite knot, using the theorem of Van Kampen. Then, we use this calculation in order to prove the new result that a crossing is strongly

essential in a composite diagram D1#D2 if and only if it is strongly essential either

in D1 or D2 (Theorem 2.38). Next, we prove that a secant in a separating sphere of a composite knot is non-peripheral but not strongly essential (Theorem 2.40). As a consequence, we prove that nugatory crossings of type II are non-peripheral but not strongly essential (Corollary 2.41).

In Section 2.4 we analyze the essentiality of a crossing in relation to the smoothings 4 Introduction

of that crossing. We look ﬁrst at the essentiality of a crossing when we smooth it according to the orientation of the diagram (Proposition 2.46). Then, we analyze what happens when we switch a crossing. We prove that if switching a crossing c transforms the knot K in the knot K0 (and the crossing c to the crossing c0) and the fundamental groups of these two knots are not isomorphic, then at least one of the crossings c and c0 must be strongly essential (Proposition 2.47). In particular, if switching a crossing c of a diagram of a non-trivial knot leads to the unknot, then c is strongly essential (Corollary 2.48).

• In Chapter3 we prove our main result: In a minimal diagram of an alternating knot all the crossings are strongly essential (Theorem 3.45). Since alternating knots can be decomposed into prime pieces, ﬁrst we focus on prime alternating knots. These are known to be of two types: hyperbolic knots and (2, p)-torus knots.

In Section 3.1 we ﬁrst introduce some basic well-known notions of hyperbolic geometry and hyperbolic knots. Then we present a more recent result by Adams [3] that will be crucial for our analysis.

In Subsection 3.1.4 we apply speciﬁc results from hyperbolic geometry (in particular the ones by Adams), and ﬁnd an unexpectedly easy solution to the proof of the new result that in a minimal diagram of an alternating hyperbolic knot all crossings are strongly essential (Theorem 3.41).

In Section 3.2 we prove that in a minimal diagram of a (2, p)-torus knots all crossings are strongly essential (Theorem 3.43). To do so we use the combinatorial structure of these diagrams.

In Section 3.3 we consider also the case of composite knots and prove our new result that in any minimal diagram of an alternating knot all crossings are strongly essential (Theorem 3.45).

• In Chapter4 we strengthen the definition of strongly essential secant to define the new concept of n-essential secant (Definition 4.13). To do so we use the cyclic 3 branched covers of S over a knot, both geometrically and algebraically. In Section 4.1 we introduce the well-known construction of covers and branched Outline 5

covers and oﬀer some examples of them.

In Section 4.2 we introduce the deﬁnition of n-essentiality. Moreover, we prove the following hierarchy: n-essential implies strongly essential (Proposition 4.17) and kn-essential for all n and k in N (Proposition 4.20). We prove that for all n ∈ N, there are secants that are strongly essential but not n-essential (Proposition 4.18).

In Section 4.3 we focus on the double branched cover, beginning with a general and known characterization. Afterwards, we prove that we can form a generator set for the fundamental group of the double branched cover along a knot from a diagram of the knot, taking a generator for each crossing (Corollary 4.27).

In Section 4.4 we use a combinatorial approach to analyze the 2-essentiality of crossings in knot diagrams. We prove that in any diagram of a non-trivial knot there must be at least three 2-essential crossings and that this limit is sharp (Theorem 4.42).

• In Chapter5 we focus on 2-essentiality analyzing the crossings of particular diagrams of diﬀerent classes of knots.

In Section 5.1 we focus on rational knots. Subsection 5.1.1 is a review on the Conway normal form for rational knots, which is used to establish a notation to work with this kind of knots and to recognize the famous correspondence with continued fractions. In Subsection 5.1.2 we prove the new result that any crossing of a minimal diagram of a rational knot is 2-essential (Theorem 5.10). Since rational knots are prime and alternating, we already knew (by Lemma 3.44) that all the crossings in a minimal diagram of any of them are strongly essential. Thus the new result here strengthens this last fact, since strong essentiality is weaker than 2-essentiality.

In Section 5.2 we analyze torus knots of type T (2, q), T (3, q) and T (5, q), for any q. We prove the new result that all the crossings of a standard braid diagram of any of these knots are 2-essential (Theorem 5.13). Note that except for the ones of type T (2, q), torus knots are non-alternating. 6 Introduction

0.3 Acknowledgements

First of all I thank my supervisor John M. Sullivan who has been supporting me during these years and made this achievement possible. I admire his enthusiasm for mathematics and his way of thinking about it. He gave me innumerable new ideas and has conveyed to me some of his beautiful mathematical visions. Moreover, he had the patience to guide me through this research project and helping me throughout. I thank Bruno Benedetti who corrected a draft of this work, and Benjamin Kutschan for sharing the office with me and helping me with the German formalities. Special thanks to Brandon Beck for language corrections and especially for standing by me in these last two crazy months. Thanks again for coming to Berlin and sorry for my anxiety and lack of time in this long and cold winter. I want also to thank the staff of the Berlin Mathematical School. They helped me with all the paper work and the formalities making my stay here in Berlin much easier. Thanks to all the mathematicians that I met in conferences and with whom I had stim- ulating conversation and gained new ideas. To list a few, Colin Adams, Paolo Bellingeri, Jason Cantarella, Elisabeth Denne, Robert Kusner and Agnese Telloni. Thanks also to the mathematicians of my group here at the Technische Universität. I also warmly thank Patrick Popescu-Pampu, whose course in algebraic topology made me first eager to learn more about topology. He also supervised my Master Thesis and introduced me to the world of mathematical research, he has been a guide in that sense. Then, I would like also to warmly thank the philosophers that I met and made me discover how fascinating knot theory can be also from their point of view. In particular I thank Marcus Giaquinto, Valeria Giardino, Øystein Linnebo and Achille Varzi. Many thanks to my friend Andrei Dorman, whose encouragement and friendship helped me to start and carry on this long journey. Thanks to my great friends Giulio Bernardi, Savio Dimatteo and Elisa Cortesi who sustained me with their presence in the hard parts of these years and shared the good moments too. Last, I warmly thank my family. They helped me throughout my life, and also in these years. Without their support I would have never been able to do this. I am truly Acknowledgements 7 lucky to have such amazing and loving parents and I wish to thank them for all they did for me, notwithstanding the knots. 8 Introduction Chapter 1

Mathematical Knots

1.1 Knots and Knot Types

1 Deﬁnition 1.1. A knot is an embedding of a circle S into the 3-dimensional Euclidean 3 space R . Equivalently, a knot is a simple (i.e. with no self intersection) closed curve in 3 R .

3 3 We will always consider the space R to be inside S , the 3-sphere. We do this by 3 3 3 considering the 3-sphere as the one point compactiﬁcation of R : S ' R ∪ {∞}, so we 3 3 have R ⊂ S . This is useful because it allows us to consider the complement space of 3 the knot in S .

Note that in other contexts knots can be deﬁned as embeddings in other 3-dimensional 3 3 spaces (like S or lens spaces) but we will always consider knots in R .

See examples of knots in Figure 1.1(a).

Deﬁnition 1.2. A link of n components is a embedding of the disjoint union of n circles 3 into the 3-dimensional Euclidean space R , that is a collection of knots (Figure 1.1(b)).

9 10 Mathematical Knots

(a) (b)

Figure 1.1: Knots and links

A knot can be viewed as a link of just one component. In the following we will focus almost exclusively on knots.

Remark 1.3 (knot/link). In some contexts it is convenient to treat knots as links (of one component), in order to gain more uniformity. Sometimes in these cases, the term knot is actually used to refer to links, the number of components is speciﬁed only when needed. For example, it is common to refer to rational knots or rational links of two components as rational knots (see Section 5.1).

Remark 1.4 (Intrinsic/Extrinsic). We begin remarking that knots are not only curves, but curves in space (i.e. embeddings of circles into a 3-dimensional space). From an intrinsic point of view, all knots are topologically equivalent, as they are all homeomorphic to 1 S . Imagine two beings “living” in two knots and walking through them as if they were walking in a gallery. Even if they can communicate, it would be impossible for them to know if they are walking in the same knot or in two different knots: They could never find if their “universe” is the same or not. To imagine a knot intrinsically does not means to imagine it as a knot in space, but as the space, which means as a possible universe. On the contrary, from an extrinsic point of view, knots are topologically distinguishable: It is for this reason, that a theory of knots becomes possible, not being about a single knot, but infinitely many.

1.1.1 Equivalences

Definition 1.1 alone is not enough to determine which knots are equivalent. For this reason, we have to define the modifications that are allowed on them, which would keep Knots and Knot Types 11 their “type” unchanged. The intention of this is to capture their “knottedness” and not their particular geometric shape. This is done by focusing on topological properties of the knot, which remain unchanged under continuous deformations like stretching or translating. The main idea is to follow our physical intuitions: We generally consider that two physical knots are the same if one can be transformed into the other without cutting and re-pasting (Gordian moves are not allowed1). In order to give a rigorous characterization of these transformations we need to to define ambient isotopies:

Deﬁnition 1.5. Let M be a manifold and N1,N2 ⊂ M. We say that N1 and N2 are ambient isotopic if there exists a continuous map h : M × [0, 1] → M, with ht(x) := h(x, t), such that:

1. ht : M → M are homeomorphisms for all t,

2. h0 = id,

3. h1(N1) = N2.

3 1 In particular, we will consider the case where M ' S and N ' S .

Deﬁnition 1.6. Two knots K1 and K2 are equivalent, and thus of the same type, if they are ambient isotopic.

This equivalence relation can be extended straightforwardly to links.

Deﬁnition 1.7. Two links L1 and L2 are equivalent, if they are ambient isotopic.

We have to note that this equivalence does not respect the “matching” of the link components, if we want that the labelling of the links are preserved we must deﬁne a stronger relation.

Remark 1.8. We need the notion of ambient isotopy in order to avoid the rise of a transformation which is too loose. For example, as Cromwell remarks in [14, p. 4], if 1 3 0 instead we just took an isotopy h : S × [0, 1] → S such that h0 = K and h1 = K , then we could transform each knot into the unknot just by pulling it tight (see Figure 1.2).

1That is, it is not allowed to cut the knot (and then paste together the loose ends). According to the famous legend, Alexander the Great cut with his sword the Gordian knot in order to untie it. 12 Mathematical Knots

Figure 1.2: Pulling a knot tight

One of the major problems in knot theory is to distinguish and classify equivalence classes of knots. The ﬁrst step in solving this problem is deciding whether two knots are equivalent.

Remark 1.9 (knot/knot type). It is interesting to notice that in the jargon of knot theory, if two knots are equivalent, we say that they are the same. Even more, there is an ambiguity of terms: Knot is sometimes used for the equivalence class, thus for knot type; and sometimes just for a representative of it. However, this is not a problem and it causes in practice no confusion, as explicitly stated for example by Lickorish [34, pag. 2], but it raises interesting questions about the methodology of knot theory. Note that at this point we already found two ambiguous ways in which the term knot is used: knot/knot type and knot/link (Remark 1.3).

1.1.2 Tame and Wild Knots

The vast majority of research in knot theory considers exclusively the so-called tame knots, which are knots that can be presented as polygonal curves. Intuitively, tame knots are abstractions of physically realizable knots because they admit a diagram with a ﬁnite number of crossings. All the theory which we will develop in the following is restricted to tame knots.

Deﬁnition 1.10. A polygonal curve is a curve in space formed by the union of a ﬁnite number of straight segments, called edges, which meet at their endpoints, called vertices.

Deﬁnition 1.11. A polygonal knot is a knot which is a polygonal simple closed curve (see Figure 1.3). Knots and Knot Types 13

Figure 1.3: A polygonal knot

Deﬁnition 1.12. A knot is tame if it is equivalent to a polygonal knot, otherwise it is wild.

Given this deﬁnition, it seems a great restriction to choose to deal with only tame knots, but this is not the case, considering that all smooth knots are tame:

Deﬁnition 1.13. A smooth knot is a knot given by a smooth (C1) function.

A proof of the following can be found in [15, p. 147]:

Lemma 1.14. A knot is tame if it is equivalent to a smooth knot, otherwise it is wild.

Wild knots are “monsters” in Lakatos terminology [33]: most of the theorem and tools used in knot theory cannot be applied to them.

Figure 1.4: A wild knot

Remark 1.15 (knot/tame knot). A third use of the term knot is for tame knot or smooth knot. 14 Mathematical Knots

From now on, a knot is a smooth knot, that is, we exclude completely wild knots from our study. Other authors (see for example [34]) prefer instead to choose the PL- catergory, (Piecewise Linear), but this is equivalent to the polygonal category and so for our purpose (by Lemma 1.14) it is equivalent to smooth as well.

1.2 Knot Diagrams

In this section we introduce the most common way to represent knots. We have already seen pictures of knots, which correspond to mathematical knots.

In order to present a knot we can choose diﬀerent solutions. For example, if we deal with polygonal or PL knots we can explicitly give coordinates of its vertices. Smooth knots can be presented via a smooth function.

However, in practice it is common to draw them according to certain conventions. The ﬁrst, most direct way to present a (tame) knot is to make a model (or picture) of a physical version of it: that is, an illustration. Figure 1.5 depicts two illustrations of two knots of the same type.

(a) (b)

Figure 1.5: Two unknots

The illustrations in Figure 1.5 are fairly clear, since it is easy to extract from them the information which is relevant for knot theorist; which we will explicitly list below. However, this is not always the case. Take for example another illustration of the same physical object, like the one in Figure 1.6. Knot Diagrams 15

Figure 1.6: A bad illustration of the unknot

This is actually an illustration of the same 3-dimensional object pictured in Figure 1.5(b), and thus of the same (geometric) knot (not only of an equivalent one). We could think of it as a picture taken from another angle, which this time hides the relevant information. These first representations, despite following some general standards (for example shadows are drawn in order to suggest 3-dimensionality), are not highly con- strained: the result can be very confusing. Illustrations are a first access to knots, but should be considered as no more than informal ways of pointing at them. Because of these inconveniences, it is necessary to define explicit conventions. To provide a more controlled representation of knots, knot diagrams are introduced. In order to draw a diagram of a knot, we project it on a sphere in an “allowed” direction, and keep track for each crossing in the projection of which strand passes over and which under. Actually, there is no need to limit ourselves to the sphere, it is also possible to project the knot onto another 2-dimensional manifold (like the plane or a torus). Nevertheless, for our purposes it is more convenient to use the sphere (Remark 1.23). Let us see how to construct a diagram from a knot.

Deﬁnition 1.16. Let K be a knot. Consider a projection (i.e. a surjective linear map) 3 2 p : R → S . The knot projection is the image of p restricted to the knot: p(K). Let x ∈ p(L) be a point in the image of the knot, this is a regular point if p−1(x) is only a point and a singular point otherwise.

Deﬁnition 1.17. A knot projection is regular if it has just a ﬁnite number of singular points and each of them is a transverse double point, that is, for every such singular point x, p−1(x) contains two points and the intersection is not tangent. 16 Mathematical Knots

Theorem 1.18. Every smooth knot admits a smooth regular projection.

Proof. For a proof see [14, p. 52].

Deﬁnition 1.19. Let K be a knot. A knot diagram of K is a smooth regular projection where, for each double point, it is given local information about the relative height of the two points.

For each crossing of the projection, we keep track of which strand underpasses and which overpasses, in this way we are able to reconstruct a knot equivalent to the original one from the diagram. The limitation to regular projections prevents the rise of messy representations, like the one in Figure 1.7.

Figure 1.7: A messy projection

Corollary 1.20. Every tame knot admits a knot diagram.

Proof. From a regular projection, whose existence is granted by Theorem 1.18, we only need to add the relative height information at each double point.

Remark 1.21. Unlike illustrations, knot diagrams are mathematical objects in themselves: They have the twofold role of representing the mathematical knot (a curve in space) and of being mathematical objects in themselves (images of functions with extra information).

As for knots, also between diagrams we deﬁne an equivalence relation via ambient isotopies:

Deﬁnition 1.22. Two knot diagrams D1 and D2 are equivalent if they are ambient 2 isotopic in the sphere S . Knot Diagrams 17

As we saw in Definition 1.5, it is important in order to well define an ambient isotopy, to be in the appropriate ambient space. In the case of knot diagrams this is the two- 2 3 dimensional sphere S and in the case of knots is the three-dimensional space R . Remark 1.23. By the previous definition the diagrams in Figure 1.8 are equivalent.

Figure 1.8: Three equivalent diagrams

Moreover, these diagrams are also equivalent to the diagram in Figure 1.9. Never- theless, if we considered diagrams on a plane rather than on a sphere, then this would no longer be the case. The diﬀerence is that on the plane the ∞-region has changed: In Figure 1.8 it is 2-sided and in Figure 1.9 it is 3-sided, but on the sphere, which is compact, there is no inﬁnite region. That is why we generally prefer to work with diagrams on the sphere. Note that the corresponding knot type is the same in each of these four diagrams.

Figure 1.9: Another equivalent diagram

In practice, knot diagrams are almost always considered up to this relation and therefore the word knot diagram normally means an equivalence class of knot diagrams. We will adopt this terminology. Several diagrams represent the same knot (like the two in Figure 1.10), and they can all be related. Figure 1.10(a) is a diagram of the previous “nasty” unknot (Fig. 1.5(b)), whose direction of projection is the one orthogonal to the paper. 18 Mathematical Knots

(a) (b)

Figure 1.10: Two non-trivial diagrams of the trivial knot

1.2.1 Reidemeister Moves

We can deﬁne a series of moves on knot diagrams which allow us to connect diﬀerent diagrams representing the same knot type: the three Reidemeister Moves. These allow us to alter the knot diagram locally in the ways depicted in Figure 1.11, they are called Reidemesiter move of type I, II and III.

(a) (b) (c)

Figure 1.11: The three Reidemeister moves

To convince ourselves that these type of modiﬁcations on diagrams do not alter the underlying knot is fairly easy, just by means of imagining a corresponding “physical” knot. Moreover these moves suﬃce to connect any two given diagrams corresponding to equivalent knots:

Theorem 1.24 (Reidemeister). Any two knot diagrams corresponding to equivalent knots can be connected by a ﬁnite number of Reidemeister moves and ambient isotopies.

Reidemeister in 1927 [51] proved his theorem using the projections of particular “triangular moves" in 3-space for piecewise linear knots. As we already remarked, this is no loss of generality. He exploited the fact that two PL knots in 3-space are ambient isotopic Knot Diagrams 19 if and only if it is possible to connect them by a series of triangular moves, which consist of replacing an edge by two edges, or viceversa. He then studied how knot diagrams changed according to these 3-dimensional triangular modiﬁcations and “translated” them into moves on diagrams. In the following it will be important to recognize minimal diagrams, that is, diagrams which present the least possible number of crossings for a given knot type. A ﬁrst characterization of diagrams is whether they are reduced or not.

Deﬁnition 1.25. A crossing in a knot diagram is nugatory if there is a circle in the sphere of the diagram that intersects the diagram transversally at that crossing and does not have any other intersection points with the diagram, see Figure 1.12. Nugatory crossings such that one of the components in which the knot is divided is unknotted, are called nugatory crossing of type I, see Figure 1.12(a). The others are nugatory crossings of type II, see Figure 1.12(b). A diagram is reduced if it does not have any nugatory crossings.

(a) (b)

Figure 1.12: Nugatory Crossings

1.2.2 Orientations

It is common to assign an orientation to knots, and thus to their projections. Oriented knots can be very useful in order to do certain operations, like the connected sum of knots (see Section 1.3.1). The crossings of an oriented knot diagram can be divided into positive and negative. The positive crossings are the ones in which the upper-strand is oriented to the right of 20 Mathematical Knots the under-strand, the strands of the negative crossings present the reverse orientation, see Figure 1.13:

+1 -1

Figure 1.13: A positive crossing (left) and a negative crossing (right)

Deﬁnition 1.26. A knot is called invertible if it is equivalent to the knot obtained from itself by reversing its orientation.

The knot 817 is the most simple non-invertible knot.

Figure 1.14: The knot 817

1.3 Knot Invariants

The main approach to the problem of determining whether two knots are equivalent is to associate to them invariants, i.e. some mathematical elements (e.g. numbers or algebraic structures) which depend only on their knot type. The general idea is that we can associate algebraic structures (groups, polynomials, etc.) to knots in such a way that equivalent knots will correspond to isomorphic algebraic structures (this being a general principle of algebraic topology). Often invariants are associated to knots diagrammatically. To do so, we start by deﬁn- ing a property on a diagram and then make sure that it is preserved under Reidemeister moves; this is one of the main applications of Reidemeister’s theorem. Knot Invariants 21

Deﬁnition 1.27. A knot diagram is alternating if, when travelling along it straight through the crossings (that is actually following the knot), in one direction or the other, we encounter alternatively over- and under-crossings (See Figure 1.15).

Figure 1.15: An alternating diagram (left) and a non-alternating diagram (right)

Deﬁnition 1.28. A knot is alternating if it admits an alternating diagram.

Note that being alternating does not depend on the particular diagram, but only on the equivalence class of the knot. Actually, in the nineteenth century it was commonly believed that all knots were alternating. This is mainly due to the fact that determining if a knot is alternating is not always easy: If we do not ﬁnd an alternating diagram, maybe it does not exist but maybe we are just not looking well enough.2 Also, even the “simplest” prime non-alternating knot, the knot 819, does not admit any diagram with less than 8 crossings (Figure 1.16).

Figure 1.16: The knot 819

Deﬁnition 1.29. The crossing number of a knot diagram is the number of its crossings.

2Using the Jones polynomial, which is a knot invariant, this has become an easy task. 22 Mathematical Knots

Deﬁnition 1.30. The crossing number of a knot is the minimal number of crossings present in a diagram of it.

In a similar fashion, we deﬁne ﬁrst the unknotting number on diagrams and then take the minimum over all diagrams.

Deﬁnition 1.31. The unknotting number of a knot diagram is the minimal number of crossing changes (that is making the under-strand become the upper-strand, see Figure 1.17) we have to perform in order to transform it into a diagram of the unknot.

Deﬁnition 1.32. The unknotting number of a knot type is the minimum over the unknotting numbers of its diagrams.

Figure 1.17: Crossing change

3 Deﬁnition 1.33. The linking number of two disjoint oriented knots K1 and K2 in S is the intersection number between the ﬁrst and a surface spanned by the second. Such a 3 surface exists since both K1 and K2 are null-homologous in S (this comes from the fact 3 that H1(S ; Z) = 0). Therefore, we have:

lk(K1,K2) = Σ · K2 where ∂Σ = K1.

This is well-deﬁned because it does not depend on the particular surface chosen. 0 0 0 Indeed, suppose ∂Σ = ∂Σ = K1. We have (Σ · K2) − (Σ · K2) = (Σ − Σ ) · K2, but 0 0 [Σ − Σ ] = 0, therefore (Σ − Σ ) · K2 = 0. Remark 1.34. The following properties of the linking number follow from the deﬁnition: lk(K1,K2) = lk(K2,K1) and lk(−K1,K2) = −lk(K1,K2). Remark 1.35 (An easy way to calculate the linking number of two knots). Considering the projections on the same plane of two oriented knots, K1 and K2, being careful that Knot Invariants 23 their intersections do not coincide with the crossing points of either of them, we can count how many times they cross each other and add +1 for every positive crossing and −1 for every negative one.

Then we will have:

lk(K1,K2) = Σ(signs of the crossing points)/2

3 3 Given a knot K in R ⊂ S , we deﬁne:

3 3 Deﬁnition 1.36. The complement of K in S is S \ K.

A tubular neighborhood of K, N(K) is K × D, where D is a disk. It is homeomorphic ◦ 1 1 to a solid torus S × D, its interior N(K) is homomorphic to a circle S times an open disk. ◦ 3 3 The exterior of K in S , which we will denote with X, is S \ N(K). Its boundary is 2 a torus T .

By the previous deﬁnition, the knot complement is a non-compact manifold without boundary, whereas the knot exterior is a compact manifold with a (2-dimensional) torus as its boundary (for a n-component link, the boundary of the exterior would have n toroidal components).

Proposition 1.37. There exists a homeomorphism that sends the interior of the knot exterior to the knot complement.

3 Proof. First, we deﬁne a map from the knot exterior to S that makes the boundary 3 torus of X collapse to the knot: f : X → S such that ∂X 7→ K. We take the tubular neighborhood N(K) to be inside a bigger tubular neighborhood N 0(K). Then, every point outside N 0(K) is mapped to itself: for x ∈ X \ N 0(K), f(x) = x, the boundary of N(K) is mapped to K and the (one side) open annulus N 0(K) \ N(K) is stretched to a punctured disk N 0(K) \ K, see Figure 1.18. 24 Mathematical Knots

N'(K) N(K) K

3 Figure 1.18: The map f : X → S

To give a geometric description, we can look at the normal disks forming the tubular neighborhoods corresponding to each point of K (like the ones in Figure 1.18). Then we assign radius 1 to the ones relative to N(K) and radius 2 to the ones relative to N 0(K). With this setting, N(K) is a tubular neighborhood of radius 1 and N 0(K) of radius 2. We assign coordinates: (k, α, r) ∈ N 0(K), where k ∈ K, α ∈ [0, 2π) and r ∈ [0, 2]. ◦ 3 0 Each point of S \ N(K) outside N (K) is mapped to itself. For each point (k, α, r) ∈ ◦ N 0(K)\ N(K) we have f(k, α, r) = (k, α, 2r − 2). Now, the restriction of f to the interior of X is a homeomorphism from the interior of the knot exterior to the knot complement.

Remark 1.38. The knot exterior is a compactification of the knot complement, of course a more straightforward compactification would just consist of filling in the missing knot, but in doing so, we would obtain the 3-sphere and thus lose any trace of the knot. On the contrary, by adding a boundary torus, we keep all the relevant information while constructing a compact space.

It is well-known that both the exterior and the complement spaces of a knot are invariants for knot types:

Proposition 1.39. Equivalent knots have homeomorphic complement and exterior spaces.

Proof. By deﬁnition, if two knots K1 and K2 are equivalent, there exists an ambient isotopy transforming one into the other. In particular, there exists a homeomorphism, h1 in the notation of Deﬁnition 1.5, of the 3-sphere such that h1(K1) = K2. Thus 3 3 S \ K1 ' S \ K2 and the same holds for the exterior space. Knot Invariants 25

1.3.1 Prime and Composite Knots

3 Deﬁnition 1.40. A ball-arc pair is a pair (B, a), where B is a 3-ball D (whose boundary 3 2 is therefore a sphere: ∂D = S ) and a is an arc such that a ⊂ B and a ∩ ∂B = ∂a.

2 3 Deﬁnition 1.41. A knot K is composite if there exists a 2-sphere S in S such that its intersection with K is exactly two points and the two portions in which K is divided are 2 3 both non-trivial. That is, S divides S into two non-trivial ball-arc pairs. This sphere is then a separating sphere. Otherwise K is prime.

There is an analogous deﬁnition for diagrams:

Deﬁnition 1.42. A knot diagram D is diagrammatically composite if there exists a circle 1 S in the plane of the diagram such that its intersection with K is exactly two points and the two portions in which D is divided are both non-trivial. Otherwise D is diagrammatically prime.

Figure 1.19: A composite knot diagram

A composite knot can have a prime diagram, however this is not the case for minimal diagrams of alternating knots. This is a result by Menasco and Thistlethwaite [38]:

Theorem 1.43 (Menasco-Thistlethwaite). Let K be an alternating knot and D(K) a minimal diagram of it. Then K is prime if and only if D(K) is diagrammatically prime.

Schubert proved in 1949 [53] a decomposition theorem into prime factors for knots: 26 Mathematical Knots

Theorem 1.44 (Schubert). There is a bijection between knots modulo equivalence and natural numbers that sends prime numbers to prime knots and preserves multiplication. In particular, every knot can be decomposed into prime knots. Moreover, this decomposition is unique up to reordering.

By the previous theorem and the result by Menasco and Thistlethwaite (Theorem 1.43), we have the following:

Proposition 1.45. Let D be a minimal diagram of an alternating knot. Then there is a decomposition of D into minimal diagrams of prime alternating knots.

1.3.2 Classes of Knots

In this section we introduce diﬀerent categories of knots. According to Thurston, all knots can be partitioned into torus knots, satellite knots and hyperbolic knots. We state in this section some well-known general facts about these three classes of knots.

Deﬁnition 1.46. A knot K is a torus knot if it is equivalent to a knot which lies on a torus.

Since the fundamental group of a torus is Z×Z, and generators are given by a meridian and a longitude (see the Appendix), every closed curve on a torus can be described by two numbers p and q, which correspond to the number of meridian and longitudinal wrappings. If the curve has no self-intersections, then p and q are relatively prime. That is why torus knots are classiﬁed via a couple of relatively prime integers and are normally referred to with expressions such as: T (p, q). We assume that p < q to have a standard labelling.

Figure 1.20: The knot T (3, 5) Knot Invariants 27

Only a speciﬁc type of torus knots are alternating, see [2, Ch. 5]. In fact, we have:

Proposition 1.47. The knot T (p, q) is alternating if and only if p = 2.

Deﬁnition 1.48. A knot is a satellite knot if it contains an incompressible, non-boundary parallel torus in its complement.

A torus embedded in the exterior X of a knot is incompressible if its fundamental group injects in the fundamental group of M, see Deﬁnition 3.23. A torus embedded in the exterior X of a knot is boundary parallel if it can be isotoped to the boundary of X, which in fact is a torus, otherwise it is non-boundary parallel.

In order to construct a satellite knot, we start with two knots: Let K1 be a knot inside a solid torus but not in a 3-ball in it and not equivalent to the core of the solid torus and let K2 be another non-trivial knot. Then, tie this solid torus in the form of

K2. The image of K1 after this modiﬁcation is another knot, K3 which is a satellite knot, whose companion knot is K2, see Figure 1.21.

(a) (b)

Figure 1.21: Satellite knots

Remark 1.49. All composite knots are satellite knots. In fact, we can take as K2 one of the two components and as K1 the other, so that K1 would be localized in a 3-ball except one arc inside the solid torus shaped as K2, see Figure 1.21(b). The following is a consequence of a result by Menasco, see [37]:

Proposition 1.50. Prime satellite knots are non-alternating.

Deﬁnition 1.51. A knot K is a hyperbolic knot if its complement is a complete hyperbolic manifold. 28 Mathematical Knots

The complement of a hyperbolic knot is a non-compact manifold, but still has ﬁnite volume [2, Ch. 5]. For more about hyperbolic manifolds, see Section 3.1. The following theorem classiﬁes knots. It is a result by William Thurston. See [58, Ch. 1].

Theorem 1.52. Any knot is either a torus knot, a satellite knot or a hyperbolic knot.

For prime alternating knots, using the previous theorem and Propositions 1.50 and 1.47, we have the following, see [37]:

Proposition 1.53. All prime alternating knots which are not (2, q)-torus knots are hyperbolic. Chapter 2

Essential and Strongly Essential Secants

2.1 Deﬁnitions

In this section we introduce various degrees of essentiality for secants of knots. A knot K is a smooth closed simple curve in the 3-dimensional Euclidean space f : 1 3 3 3 3 3 S → R . As already mentioned, we consider R inside the 3-sphere: R ⊂ S ' R ∪{∞}. This is useful because it allows us to work with the exterior and the complement of K 3 in S . For the purpose of this research, we will only consider tame oriented geometric knots in space. A secant is a straight segment connecting two points of the knot. The first study on topologically essential and inessential secants was made by Kuper- berg in the context of investigating the presence of secants, trisecants and quadrisecants in a tame knot. Multi-secants had already been investigated, first by Pannwitz [47] and later by Morton and Mond [41]. Later on, Kuperberg [32], while proving the existence 3 of quadrisecant for every tame knot in R , introduced the concept of essential secants (in his words topologically non-trivial). More recently, this concept was studied and developed by Denne in her Doctoral Thesis [19]. Moreover, Denne, Diao and Sullivan [20] used it to find lower bounds for the ropelength of non-trivial knots. Essential secants were also used by Denne and Sullivan to study the distortion of knotted curves [21]. Here we will adopt the terminology introduced by Denne, using equivalent definitions.

29 30 Essential and Strongly Essential Secants

We approach the problems related to the secants of knots by working not only with secants, but also with secant arcs, which are proper arcs in the knot exterior. We introduce a new degree of essentiality between what has previously been defined (stronger than essential but weaker than strongly essential): non-peripherality (Definition 2.5). Let us start by defining what secants are. We use the notation introduced in the previous chapter: Let K be a knot and N(K) a tubular neighborhood of it: N(K) ' ◦ 2 2 1 3 3 D × K ' D × S . Let X be the exterior of K in S : X = S \ N(K). The boundary of X is a torus: ∂N(K). 3 Consider the map of Proposition 1.37 from the knot exterior f : X → S that makes ∂X = ∂N(K) collapse to K. Recall that this is the map which sends the interior of the knot exterior homeomorphically to the knot complement.

Deﬁnition 2.1. Let K be a knot and a and b two distinct points on it. A secant-line is a straight oriented line intersecting K in at least two points. A (straight) secant connecting a to b, S or ab, is a portion of a secant-line with endpoints on the knot and no internal intersection with it. A secant arc, A, relative to a straight secant S is an embedded arc in the knot exterior that maps under f to an arc homotopic with ﬁxed endpoints to S. We have that for any −1 1 −1 point x ∈ K, f (x) = S ∈ ∂X and ∂A ∈ f (∂S).

A s b

Figure 2.1: A secant arc corresponding to the secant S

Remark 2.2. By the previous definition we have that if A1 and A2 are two secant arcs relative to a secant S, then f(A1) is homotopic with fixed endpoints to f(A2). This is not necessarily the case for A1 and A2, in fact they might not even have the same endpoints. Definitions 31

Remark 2.3. Not all pairs of points in the knot give rise to a straight secant. By the previous deﬁnition, we exclude those which would individuate a tri- or multi-secant (portion of secant-lines intersecting the knots in more than 2 points). However, a secant (as well as a secant arc) can be identiﬁed by two points on the knot.

We deﬁne now peripheralality, to do so, we need the following notions. The boundary 2 of the knot exterior is a torus embedded in the knot exterior: i : T → X. If we look 2 at the map induced in homotophy we have the following injection: i∗ : π1(T ) ,→ π1(X). 2 Since π1(T ) ' Z × Z, we can always identify the subgroup of π1(X) isomorphic to Z × Z and corresponding to the boundary torus. This group is generated by a meridian and a longitude of the boundary torus (see Section 5.2.1)

Deﬁnition 2.4. Let K be a knot and X its exterior, a peripheral subgroup is a subgroup isomorphic to Z×Z of π(X) which is a conjugate to a subgroup of the group corresponding to the fundamental group of the boundary torus.

Definition 2.5. A secant arc A corresponding to the secant S is peripheral if it is homotopic as a curve with fixed endpoints to a curve in the boundary torus of the knot exterior. Otherwise, we say that A is non-peripheral. We adopt the same definition for secants as well.

Note that this notion is well-deﬁned: By deﬁnition if a secant arc relative to a secant is peripheral, all secant arcs relative to that secant will be so. In fact as we have seen in

Remark 2.2, two secant arcs A1 and A2 relative to the same secant, might not be homotopic as paths with ﬁxed endpoints, but they can only diﬀer by a peripheral component:

We have that A1 is homotopic to A2 united to an arc that goes around the meridional of the boundary torus a certain fractional number of times. We now introduce a new notation in order to deﬁne essential and strongly essential secants. Let K be a knot and S = ab one of its secants: It corresponds to a portion of secant line and it divides the (oriented) knot K into two arcs, K1 from a to b and K2 from b to a. We have associated secant arcs to the secant S, let A be such a secant arc, and let a˜ and ˜b be its endpoints. These points are on ∂X and are in the circles identiﬁed by a and b, the corresponding endpoints of the secant. 32 Essential and Strongly Essential Secants

Now, we also consider the lift of the knot itself in the boundary knot exterior. Call k1 a lift of the arc K1 to the boundary torus such that its end points are a˜ and ˜b, and such that the loop A ∗ k1 has linking number 0 with K—note that we can concatenate these two arcs because their endpoints coincide. This loop is uniquely determined up to homotopy. In the same way, call k2 the lift of the other portion of the knot with end-points a˜ and ˜b such that lk(A ∗ −k2,K) = 0. Then, k1 ∗ k2 is a longitude of the boundary torus.

Now call the loops oriented along K, α := A ∗ k1 and β := A ∗ −k2. We take the point a˜ on A as the base-point for the fundamental group of the knot exterior π1(X), and consider the homotopy classes of α and β. Note that we choose a speciﬁc base-point for each secant arc. This is not a problem because, even if we switch the base-point, we will get an isomorphic group (even if the isomorphism is not standard but dependent on the path chosen to connect the two base-points). In the following we will be interested in checking the triviality of certain elements of these fundamental groups. This is invariant if we consider isomorphic groups or conjugates. Call µ a meridian in the boundary torus and λ one of its longitudes having linking number zero with K. We then have lk(µ, K) = 1 and lk(λ, K) = 0.

Remark 2.6. In the previous notations we have that β is homotopic to λ−1α. In fact −1 −1 −1 we have that αβ ∼ k1 ∪ k2 is a longitude, and lk(β α, K) = 0, thus αβ ∼ λ and therefore it holds that β ∼ λ−1α.

Lemma 2.7. Using this terminology, the secant arc A is peripheral if and only if the k homotopy class [αλ ] is trivial for some k ∈ Z. Equivalently we can replace α by β.

Proof. By definition we know that A is peripheral if it is homotopic with fixed endpoints to an arc in the boundary torus. But this is equivalent to saying that α = A ∪ k1 is homotopic to a loop in the boundary torus, since k1 by definition lies on it. Equivalently, m n α ∼ µ λ for some m, n ∈ Z, because any closed curve on a torus can be written as a combinations of meridians and longitudes. But [α] = [µmλn] implies that m = 0 since lk(α, K) = 0 and lk(µmλn,K) = n. That is [α] = [λn], which is equivalent to [αλk] = 0 Definitions 33 for some k ∈ Z. We can replace α by β using Remark 2.6.

Using this terminology we introduce another notion:

Deﬁnition 2.8. A secant arc A is inessential if either α or β are homotopically trivial in X, otherwise it is essential. The same terminology is also adopted for the corresponding secant.

In Figure 2.2 you can see an inessential and an essential secant of the trefoil knot.

(a) (b)

Figure 2.2: An inessential secant (a) and an essential secant (b)

Remark 2.9. As we mentioned, in order to check the essentiality of a secant arc we have to check the triviality of the class of α. This is equivalent to check the triviality of the class of any loop conjugated to it. Thus we could also check the free homotopy class of α.

The following is an equivalent deﬁnition for essential secants, given by Denne. First Denne notices that a knot with a secant forms a knotted θ-graph, composed by two vertices (the end points of the secant) and three edges:

Proposition 2.10 (Denne). Given two points a, b ∈ K we can consider the two arcs in which K is divided, K1 and K2. Given the θ-graph relative to a and b, S is inessential if and only if there is either a disk D1 bounded by K1∗S having no interior intersections with the knot K = K1 ∗ K2, or a a disk D2 bounded by S ∗ K2 having no interior intersections with the knot. These disks can have intersections with S.

Proof. The condition in the proposition is equivalent to the following, see [19] for details: 3 3 Consider the knot complement C = S \ K = S \ K1 ∗ K2, and let p1 be a parallel loop 34 Essential and Strongly Essential Secants

to K1 ∗ S in C, i.e. there is an embedded annulus co-bounded by p1 and S ∗ K1 in C, such that it is homologically trivial, i.e. lk(p1,K) = 0, see Figure 2.3.

b a

Figure 2.3: The loop p1

Then, requiring the existence of the disk D1 in the statement of the proposition is equivalent to requiring that the homotopy class of p1 is trivial in C, with appropriate choice of base-point. Of course, the same construction holds for the other side, were we would ﬁnd a loop p2 parallel to S ∗ K2.

Now, the loop p1 can be considered in the knot exterior X, because the tubular neighborhood around the knot can be chosen with arbitrary small radius it is by construction homotopic to α (in fact, by deﬁnition we have lk(α, K) = 0). The loop p2 will instead be homotopic to β.

Lemma 2.11. If a secant arc is non-peripheral, then it is essential.

k Proof. We have that A is peripheral if and only if α ∼ λ , for some k ∈ Z. Therefore β ∼ λk−1. On the other hand, A is inessential if and only if α ∼ id (and β ∼ λ−1) or α ∼ λ (and β ∼ id). Thus, inessentiality is a particular case of the peripherality, where k = 0 or k = 1.

It is possible to find examples of secants being essential but peripheral. In the previous notation, we just have to chose a k∈ / {0, 1}. For example, in Figure 2.4 you can see that the arc depicted satisfies this property; however, it is not a straight secant. In order to find a straight secant corresponding to it, we would have to reconfigure the θ-graph depicted so that the arc will become straight (of course then the knot will assume a more complicated shape). Definitions 35

Figure 2.4: An essential but peripheral arc

Deﬁnition 2.12. The secant S is simply inessential if there exists a corresponding secant arc A such that either α or β bounds an embedded disk in the knot exterior.

If A is inessential the corresponding loop α (or β) is trivial and thus bounds a disk, but this disk is not necessary embedded. In fact it can have self intersections and intersections with the arc itself. For example, the secant in Figure 2.1 is simply inessential, but the one in Figure 2.6 is not. Anyhow, this last secant is also inessential since the diagram is a diagram of the unknot and as we will see (Lemma 2.13), the unknot does not have any essential secant.

Figure 2.5: A simply inessential secants

We present now two known facts that characterize the unknot in terms of secants:

Lemma 2.13. If K is the unknot, then every secant of K is inessential.

Proof. The knot group of the unknot is Z, which is equal to the ﬁrst homology group of X. This means that a curve having linking number zero with the knot is homotopically 36 Essential and Strongly Essential Secants trivial, thus, in the previous terminology α, which needed to have linking number zero with the knot, is trivial.

Lemma 2.14. If the secant S is simply inessential, then its union with one of the two sub-arcs of the knot, between a and b (K1 or K2), forms a trivial knot.

Proof. If S is simply inessential, then so is the corresponding secant arc A; then, either α or β will bound an embedded disk. But the only knot which bounds an embedded disk 3 is the unknot, so either α or β is the unknot. So, also the corresponding arcs in S , the secant S and K1 (or K2), must form the unknot.

We can use Lemma 2.13 to present examples of inessential secants which are not simply inessential. It is enough to take a complicated version of the unknot, like the one in Figure 2.6 (which is a “double trefoil”), with the secant connecting the two double ends.

Figure 2.6: An insessential but not simply inessential secant

By the characterization of simple inessentiality given by Lemma 2.14, we can check that the secant in Figure 2.6 is not simply inessential since its union with either sub-arc of the knot is a trefoil knot, and therefore cannot be the unknot. As already done by Denne, we characterize the unknot as a knot having a secant that is inessential “from both sides”:

Proposition 2.15 (Denne). Let K be a knot, S a secant of K and A a corresponding secant arc. If both α and β are homotopically trivial, then K is the unknot.

Proof. To prove this result we will use Dehn’s Lemma (Lemma 5.14). The secant arc A belongs to X, we perturb it so that it does not have interior intersections with ∂X. Deﬁnitions 37

−1 Let D1 be a disk in X bounded by α and D2 one bounded by β (we take a loop with inverse orientation in order be able to glue these disks). By gluing them together along A = ∂D1 ∩ ∂D2, we obtain another disk, D. Note that D1 and D2 exist and that their interiors are disjoint from K since α and β are homotopically trivial.

We have that ∂D1 = A ∗ k1, where k1 is (as deﬁned before) the lift to the boundary torus of one portion of the knot and ∂D2 = −A ∗ k2.

The disk D, bounded by k1 ∗k2 = K, is not necessarily embedded but the singularities lie away from the boundary.

Thus, applying Dehn’s Lemma, we get an embedded disk bounded by the lift of K. Projecting down, we obtain that K is the unknot.

2.1.1 Strongly Essential Secants

We now introduce a stronger property for secants and secant arcs: strong essentiality. Denne [19] already introduced this concept (it was then developed by Denne, Diao and Sullivan [20]). We introduce it using an equivalent deﬁnition.

Let µ be a meridian loop of the boundary torus, which links the knot near a˜, one of the two boundary points of A. Let again α be a loop in the boundary torus with linking number zero with the knot and composed by the union of the secant arc and k1, the lift of a portion of the knot.

For a secant, we are interested in considering the homotopy class of a particular loop in the knot exterior that goes through x0 and encircles A.

Deﬁnition 2.16. The loop lA around the secant arc A, is the loop that travels along the arc A, then wraps around a meridian, travels back along A and wraps around another meridian in the opposite direction. See Figure 2.7.

Remark 2.17. The loop lA has linking number 0 with the knot. 38 Essential and Strongly Essential Secants

a b lA

Figure 2.7: A loop ‘around’ a secant-arc

Deﬁnition 2.18. The secant arc A is strongly essential if lA is non-trivial in the fundamental group of the knot exterior.

Using the commutator between the meridian and the loop α, we can give another characterization of strongly essential secants; but ﬁrst we need the following:

Proposition 2.19. Let K be a knot and A a secant arc of it. Consider µ in π1(K), a meridian loop close through the base-point a˜, and the loops α and β as elements of

π1(K). We have [µ, α] = [µ, β].

Proof. We have that α−1β is homotopic to a parallel of the knot K, more speciﬁcally, it is a longitude λ−1 which commutes with µ since the torus has abelian fundamental group: One has: [α−1β, µ] = α−1βµβ−1αµ−1 = λ−1µλµ−1 = µλ−1λµ−1 = id, where id is the identity element of π1(X, x0). Moreover we have: [µ, α][µ, β]−1 = (µαµ−1α−1)(µβµ−1β−1)−1 = (µαµ−1α−1)(βµβ−1µ−1) = µα(α−1β)µ−1µβ−1µ−1 = µββ−1µ−1 = id. And thus the result: [µ, α] = [µ, β].

Proposition 2.20. Let S be a secant of K and A the corresponding secant arc. The commutator [µ, α] is non-trivial if and only if A (and thus also the corresponding secant S) is strongly essential.

Proof. We can slide the loop [µ, α] along the portion of the knot k1 so that it encircles a meridian next to a˜ and the second meridian next to ˜b. Like this it will be clearly homotopic to the loop lA, as you can see from Figure 2.8; for more details see [19]. Deﬁnitions 39

g1 [μ, ]

g2 LS g3

Figure 2.8: Strong essentiality

Remark 2.21. Note that strong essentiality relative to “one side” (that is to the portion of the knot K1, and thus to its lift k1) is equivalent to the one relative to the other side

(that is to the portion of the knot K2 and its lift k2). This is not the case for essentiality. In other words, to check the essentiality of a secant we have to verify the non-triviality of both α and β. Here by Proposition 2.19, [µ, α] is non-trivial if and only if [µ, β] is non-trivial.

Remark 2.22. As the name suggests, strong essentiality is stronger than essentiality. In fact if a secant S is strongly essential, then [µ, α] (and thus also [µ, β]) is non-trivial, but this implies that both α and β are also non-trivial and thus that S is essential.

The notion of strong essentiality is also stronger than non-peripherality, in fact we have the following:

Proposition 2.23. If a secant arc A is strongly essential, then it is non-peripheral.

k Proof. If the secant arc A is peripheral, then αλ is homotopically trivial for some k ∈ Z, and so α ∼ λ−k. But then we have: [µ, α] = [µ, λ−k] = id since meridian and longitude commute.

Thanks to the previous proposition and Lemma 2.11, we have the following hierarchy for secants and secants arc of knots:

strongly essential =⇒ non-peripheral =⇒ essential 40 Essential and Strongly Essential Secants

We will see that these are strict inclusions, that is, that these notions are actually all distinct.

2.1.2 Wirtinger Presentation

In this section we present a well-known algorithm which allows us to compute a presentation of the knot group from a diagram D of a given knot K: the Wirtinger Presentation.

Let us denote by X the exterior of K in the 3-sphere and G := π1(X) its fundamental group. The algorithm used to compute the Wirtinger presentation consists of the following steps:

1. Choose an orientation for D. (It is needed for the computations but the choice will not inﬂuence the result.)

2. Take a generator gi for each strand. First of all we consider the base-point of the fundamental group of X to be a point above the diagram (identiﬁable with the eye of the reader). Then for each strand

gi, we choose a loop that encircles the strand in a positive way (i.e. with linking number +1 with the knot). This will be indicated in knot diagram as in the top of Figure 2.9 (left). The dotted arrows actually represent loops that meet at the base point which is situated above the plane of the diagram like in 2.9 (right).





Figure 2.9: Notation for the generators of π1(X)

  

 



 

Deﬁnitions  41

   

 

  

   Figure 2.10: Generators of π1(X)

    3. Take a relation for each crossing of D.   Each relation corresponds to the fact that we can slide a loop around two arcs meeting in a crossing to the other side of the crossing, as can be seen from the

diagrams in Figure 2.11. 





Figure 2.11: Sliding a loop through a crossing

In Figure 2.10 g1 is the generator corresponding to the left under-strand; g2 is the

one corresponding to the right under-strand and g3 is the one of the over-strand. As we can see in the two diagrams (positive and negative crossings) of Figure 2.11, we have independent of orientation:

g2g3 = g3g1

−1 For the positive crossing we have g2g3 = g3g1 which is equal to g2 = g3g1g3 . For the negative crossing we obtain the same results since from Figure 2.11 we have −1 −1 −1 g3 g2 = g1g3 which is again equivalent to g2 = g3g1g3 . 42 Essential and Strongly Essential Secants

In both cases (i.e. positive and negative crossings) every relation expresses the fact that the two under strands are conjugated by the over-strand. If we consider a digram D of K, with n crossings (and therefore n strands), we obtain the following presentation:

G = hg1, . . . , gn | r1, . . . , rni

See [52, Ch. 3] for more details.

Remark 2.24. The fundamental group of the knot exterior is isomorphic to the fundamental group of the knot complement.

Using the Wirtinger presentation, we can derive different properties of the fundamental group of a knot exterior. For example we can find relations among loops (up to homotopy) around the crossings of a region in a knot diagram. Considering the crossings of the diagram as vertices and the strands connecting them as edges we can interpret a knot diagram as a particular 4-valent graph. Then, a region of the diagram is a polygonal face of this graph. We can consider particular loops xi around the crossings of the region, all oriented in the same way and all connecting two areas outside the region, as in Figure 2.12. Each of these loops is to be connected to the common base-point, which is identified with the eye of the reader. In the figure we imagine pulling up the top parts of the loops and connecting them to the base-point situated somewhere above. With this choice it is possible to compose loops without ambiguity.

Lemma 2.25. Consider a k-sided region of D. The product x1 ··· xk of all the loops xi, i = 1, . . . , k, taken in the same direction, as in Figure 2.12, is trivial in the fundamental group of X.

Proof. For every arc deﬁning the region, we can consider an element of the Wirtinger presentation of the fundamental group of X, a loop hi which goes from outside, inside the region, for i = 1, . . . , k. Any of these loops is homotopic to a standard Wirtinger generator or its inverse. Essential Crossings 43

h2 h1 x2 xn

Figure 2.12: Region relation

−1 −1 We can then express all the xis in terms of the hi: x1 = h1h2 , x2 = h2h3 ,... In −1 general we have: xi = hihi+1 with indices modulo k. The choice of the notation allows us to directly infer our thesis:

−1 −1 −1 −1 x1 . . . xk = h1h2 h2h3 . . . hk−1hk hkh1 = id where id is the identity of π1(X).

Remark 2.26. We can use the Wirtinger presentation in order to express the commutator we have to consider in order to check if a secant is strongly essential or not. This will be helpful for calculations. Note for example that in the case of the trefoil knot in Figure 2.8, we can use the

Wirtinger presentation for π1(X) and express in terms of the Wirtinger generators the −1 −1 −1 −1 loops lA ∼ [µ, α]. We have: µ = g1 and α = g3g1 . Thus, [µ, α] = µαµ α = −1 −1 −1 −1 −1 −1 g1 g3g1 g1g1g3 = g1 g3g1g3 = g1 g2 by using the Wirtinger relation g3g1 = g2g3.

That is, in this case the meridian element µ corresponds to the Wirtinger generator g1 −1 and the commutator [µ, α] will correspond to g1 g2.

2.2 Essential Crossings

In this section we consider how to associate secants directly to crossings in a knot diagram. This allows for a combinatorial treatment of secants. 44 Essential and Strongly Essential Secants

On one hand, given a knot diagram we can consider for each crossing a corresponding secant, the straight “polar axis” (this terminology comes from Menasco [37]) connecting the under-strand to the over-strand meeting in the crossing. On the other hand, given a knot and a secant it is always possible to ﬁnd a diagram for that knot such that the considered secant is a straight secant corresponding to a crossing, that is, orthogonal to the plane of projection. Thus, given a knot K and a diagram of it D(K), we will investigate the essentiality of its crossings.

Deﬁnition 2.27. A crossing is called peripheral, essential or strongly essential if the corresponding secant is so.

In the next chapter we will prove our main result: In a minimal diagram of an alternating knot all crossings are strongly essential. This result relates the minimality of a diagram of an alternating knot to the essentiality of its crossings. Let us start here with some results about the existence of strongly essential crossings. First of all we need to introduce a notation for crossings in a knot diagram. Given a knot diagram, at each crossing three arcs converge. Recall that, as depicted in Figure 2.13, there are two kinds of crossings: positive and negative. Let I be the set of all strands of a knot diagram D, and C the set of all crossings c = (a, b1, b2) where a, b1, b2 ∈ I. In this notation a is the over-crossing and b1 and b2 are the under-crossings. The set C is divided into the set C+ of positive crossings and the set C− of negative crossings. In the ﬁrst case b1 is the under-strand coming in the crossing and b2 is the one going out. For negative crossings the roles of b1 and b2 are inverted, see Figure 2.13.

a a b2 b2

b1 b1

Figure 2.13: A positive and a negative crossing

We denote as usual by gi the Wirtinger generator corresponding to the strand i ∈ I. We use such non-standard notation for negative crossings because it allows us to have a Essential Crossings 45 uniform labelling for the loops around a crossing:

Deﬁnition 2.28. Let c = (a, b1, b2) be a crossing in a knot diagram. The loop around the crossing c is l := g g−1, that is, a loop in π (K) formed by the product of two c a b2 1 Wirtinger generators corresponding to the over-strand and to the inverse of the second under-strand. Consider the positive crossing in Figure 2.14(a). There are 8 possibilities for loops that are products of two of the three Wirtinger generators corresponding to the strands of the crossing: g g−1 = g−1g , g−1g = g g−1, g g = g g and g−1g−1 = g−1g−1. The a b2 b1 a a b1 b2 a a b2 b1 a a b1 b2 a ﬁrst four loops have linking number 0 with the knot, while the other four have linking number ±2 with the knot. (Remember that the base-point for the Wirtinger presentation of π1(X) has to be considered to be the eye of the reader). We choose as the standard loop around a crossing a loop that has linking number 0 with the knot, that is, the one that we already analyzed in order to study strong essentiality. If A is the secant arc corresponding to the secant relative to a crossing (the vertical arc connecting the over-strand and the under-strand of that crossing), then lA ∼ lc.

By convention we choose lc to be the loop with linking number zero and with the generator corresponding to the over-strand in a positive power: l = g g−1. c a b2

a g g -1 b a b2 a 2 b2

g g -1 a b2 b1 b1

(a) (b)

Figure 2.14: The loops around a positive crossing (a) and around a negative crossing (b)

For a negative crossing, as you can see in Figure 2.14, it is exactly the same situation as for positive crossings because the labels of the under-strands are switched. This allows us to have a unified notation for the loop around the crossing c as l = g g−1, c a b2 independent from the sign of the crossing. Though, as is clear from the figures, if we 46 Essential and Strongly Essential Secants orient the crossing such that the over-strand is pointing in the north-east direction (like in the previous figures), we see that in a positive crossing lc is a vertical loop, while in a negative crossing it is horizontal.

Lemma 2.29. Let D be a knot diagram and c a nugatory crossing of D of type I. Then, c is simply inessential.

Proof. A nugatory crossing of D of type I divides the knot in at least a trivial part. Consequently at least one among α and β must bound an embedded disk in the knot exterior.

Lemma 2.30. Let c = (a, b1, b2) be a crossing. Then c is not strongly essential if and only if ga ∼ gb2 , or equivalently if and only if ga ∼ gb1 , in the Wirtinger presentation of

π1(X).

Proof. We have to investigate the triviality of the loop lc around the crossing c. As before, we can write this loop in terms of the Wirtinger generators (see Figure 2.14):

l = g g−1 = g−1g c a b2 b1 a

Thus lc is homotopically trivial in the knot complement if and only if ga ∼ gb2 , if and only if ga ∼ gb1 .

Theorem 2.31. Every diagram D(K) of every non-trivial knot K has a least one strongly essential crossing.

Proof. This follows from the fact that the only knot with knot group isomorphic to Z is the unknot, which can be seen for example as a consequence of the Gordon-Luecke Theorem [25]. Consider the Wirtinger presentation of the knot group relative to the diagram D(K). By contradiction, if none of the crossings were strongly essential, then by Lemma 2.30 at any crossing all three generators would be equal. Then from the Wirtinger presentation, we would obtain a presentation of the knot group with a single generator and no relation. In this case π1(X) would be Z, the free group in one generator; and thus X would be the exterior of the unknot. Secants Under Knot Composition 47

Corollary 2.32. Every diagram D(K) of every non-trivial knot K has a least one essential crossing.

Proposition 2.33. A diagram D represent the unknot if and only of all its crossings are inessential.

Proof. By Lemma 2.13 all the secants of the unknot are inessential, and thus also all the crossings of a diagram representing it. On the other hand, if all the crossings of a diagram D are inessential, by the previous corollary, D cannot be a diagram of a non-trivial knot.

2.3 Secants Under Knot Composition

In this section we study the behaviour of essential and strongly essential secants under knot composition. In particular, we analyze in details the essentiality of the crossings of a composed knot diagram. Since we have a topological notion of knot composition and not a geometric one, when we compose a knot with another one, not all of the original secants will remain secants in the new knot. However, when we compose two diagrams, we can still consider the essentiality of the crossings as crossings of one of the summands diagrams or as crossings of the composite diagram. In fact, each crossing of D1#D2 is a crossing in either D1 or D2. To understand whether the essentiality of these crossings is preserved, we start with the well-known calculation of the knot group of K1#K2 in terms of the knot groups of

K1 and K2. To do so we apply the standard decomposition of the exterior of K1#K2 and apply the theorem of Van Kampen (Theorem 5.18). First we need the following remark:

Remark 2.34. Let K be a knot and X its exterior. Any meridian loop in π1(X) has iniﬁnte order. In fact such a loop has linking number one with the knot and thus maps to a generator of the abelianiztion H1(X) of π1(X).

Theorem 2.35. Let K1 and K2 be two knots, call K their composition. Let X denote the exterior of K and Xi the exterior of Ki, for i = 1, 2. Then π1(X) is the free product of π1(X1) and π1(X2) amalgamated over a meridian loop. 48 Essential and Strongly Essential Secants

Proof. The boundary of Xi is a torus. Then we choose the preferred meridian mi in a ◦ ◦ ◦ ◦ 1 open neighborhood of this torus: mi ⊂ ∂X× I' T× I, (where I= (0, 1) 'D ), which has linking number +1 with the knot. The base-point for π1(Xi) is picked in the preferred meridian and thus belongs to the neighborhood of the boundary of Xi (the two preferred meridians will be identified in the knot group of the composite knot K). In order to apply the theorem of Van Kampen, we define two path-connected open ◦ sets U and V such that: X' U ∪ V . Let U = X1 \ ∂X1 and V = X2 \ ∂X2. We need now to define an appropriate gluing map for X1 and X2.

For this purpose, we ﬁrst present the spaces Xi as a 3-ball minus a tube. The space

Xi is deﬁned as the 3-sphere minus a tubular neighborhood of the knot. So, we take out ◦ a 3-ball in N(K) from both the 3-sphere and the tubular neighborhood, this 3-ball is a ◦ ◦ ◦ 3 1 2 open 3-ball of the form of a cylinder D 'D × D : ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 3 3 1 2 1 2 3 1 2 Xi = S \ N(K)' (S \ D × D ) \ (N(K) \ D × D ) ' (S \ D × D ) \ ◦ ◦ ◦ ◦ ◦ ◦ 1 2 1 2 3 3 1 2 3 1 2 (S × D \ D × D ) ' (S \ D ) \ (D × D ) ' D \ (D × D ).

That is, we can see Xi as the 3-ball minus a neighborhood of a knotted arc, like in

Figure 2.15. The boundary of Xi is now divided into the boundary of the ball and the boundary of a tubular neighborhood of the knotted arc. These boundary parts intersect ◦ 0 1 2 0 2 1 2 in S × S (i.e. two circles). We have: ∂Xi ' S \ (S × D ) ∪ D × D . The choice of the open ball which has been removed, is made in such a way that the preferred meridian can be homotoped to one of the two circles where the boundary parts intersect (so that it will be possible to identify them with the gluing).

Figure 2.15: A knot exterior as a ball minus a tube

Now, in order to construct X we can glue X1 and X2 along the “external” component Secants Under Knot Composition 49

◦ 2 0 2 of their boundary, that is S \ (S × D ) in such a way as to make them overlap (we need this overlap in order to apply Van Kampen Theorem), see Figure 2.16. To do so we take ◦ ◦ 2 0 2 a neighborhood of this boundary component: S \ (S × D )× I (see Figure 2.16) and we glue them along it.

Figure 2.16: The neighborhood of the boundary to be glued

◦ ◦ ◦ ◦ 2 0 2 1 1 2 0 2 1 1 Since S \ (S × D ) ' S × D , we have: S \ (S × D )× I' S × D × I. Then we ◦ 1 1 glue them and in doing so we identify the two S × D × I.

Doing such gluing, we also identify the preferred meridians m1 and m2.

Given the two open sets: U = X1 \∂X1 and V = X2 \∂X2, we have: π1(U) = π1(X1) and π1(V ) = π1(X2). 3 Let Y = U ∪V ' X \∂X, then Y = S \N(K), that is, Y is a manifold homeomorphic 3 to the knot complement S \ K. ◦ 3 Since X ' S \ N(K) is the knot exterior, we have π1(X) = π1(Y ). Then, U ∪V ' Y , ◦ ◦ 1 1 1 1 1 where U and V are glued along S × D × I. And U ∩ V ' S × D × I' S so that

π1(U ∩ V ) = Z ' hmi. We have the following commutative diagram:

i 11 U P j1 PPq U ∩ VY P 1 Pq j i2 V 2

The induced maps on the fundamental group are the following:

π (X ) j i1∗: 1 1 XXz1∗ π (X) Z XXXz : 1 i 2∗ π1(X2) j2∗ 50 Essential and Strongly Essential Secants

Where i1∗(m) = m1 and i2∗(m) = m2. (The common base-point is in the intersection U ∪ V ). Then, by the theorem of Van Kampen we have:

∼ π1(X1) ∗ π1(X2) ∼ π1(X) = −1 = π1(X1) ∗Z π1(X2) hhi1∗(m)(i2∗(m)) ii

−1 −1 Where hhi1∗(m)(i2∗(m)) ii = hhm1m2 ii is the smallest normal subgroup containing −1 m1m2 . That is, the knot group of K1#K2 is the direct product of the knot group of the two components amalgamated over the preferred meridians (m1 and m2 are identiﬁed).

See [52, Appendix A], for the following result about the theorem of Van Kampen:

Lemma 2.36. In the theorem of Van Kampen, we have the following maps on the fundamental group induced by the inclusions:

i π (U) j1∗ 1∗: 1 XXz π1(U ∩ V ) π1(U ∪ V ) XXz : i2∗ π1(V ) j2∗

If the maps i1∗ and i2∗ are injections, then also the maps j1∗ and j2∗ are injections.

Lemma 2.37. Let K1 and K2 be two knots, call K their composition. Let X denote the exterior of K and Xi the exterior of Ki, for i = 1, 2. Then π1(X1) and π1(X2) are subgroups of π1(X).

Proof. For the fundamental groups of any knot exterior X, we have seen (Remark 2.34) that any meridian has inﬁnite order, and thus, that the group Z can be seen as a subgroup of π1(X).

That is, the maps i1∗ and i2∗ are injections, and so by the previous lemma, so are j1∗ and j2∗.

When we compose two knot diagrams, we have the additivity of crossing numbers. Nevertheless, given two minimal diagrams, we do not know if composing them will give rise to another minimal diagram. That is, cr(D1#D2) = cr(D1) + cr(D2), but cr(K1#K2) ≤ cr(K1) + cr(K2).

When we consider the strong essentiality of a crossing we analyse the loop lc around it in the knot complement X. Looking at the crossings of D1#D2, the loops we consider Secants Under Knot Composition 51

belong to the exterior of K1#K2. But actually, as we have discussed, this can be seen as the result of the gluing of X1 and X2 and the loops which were in X1 (or in X2) even considered in the manifold resulting after the gluing, can still be considered as loops in one of the components.

Theorem 2.38. Let D1 or D2 be two knot diagrams. Then, a crossing is strongly essential in D1#D2 if and only if it is strongly essential either in D1 or D2.

Proof. By Lemma 2.37, both π1(X1) and π1(X2) inject into π1(X) and thus non-trivial elements have non-trivial images. More speciﬁcally, this implies that a crossing which is strongly essential in one of the two components, will remain strongly essential as seen in the composition. Looking at the same map, it is also clear that trivial loops are sent to trivial loops.

Corollary 2.39. Let D1 and D2 be two knot diagrams. Then, the number of strongly essential crossing of D1#D2 is the sum of the number of strongly essential crossings of

D1 and D2.

However, the same statement does not hold for essential crossings. Consider for example the composite knot diagram in Figure 2.17. If we divide the diagrams into factors, we will get a minimal diagram of the trefoil and a non-minimal diagram of the same knot. In fact, the crossing in the middle of the composite diagram in no longer essential in the diagram of the trefoil knot.

Figure 2.17: A secant in a composite knot

2 Theorem 2.40. Let K be a composite knot and P ' S a separating sphere. Then any secant A that lies on P is non-peripheral but not strongly essential. 52 Essential and Strongly Essential Secants

Proof. Suppose that the secant arc A lies on the separating sphere P . In order to check if it is strongly essential, we consider the loop l := lA around it. The orientation of the two strands intersecting P must be opposite and because lk(l, K) = 0, see Figure 2.18. We can choose l to lie on P . In fact, we deﬁne l as the boundary of a tubular neighborhood of A in P .

D2 D1

Figure 2.18: “Disks" constructions

So, there are two disks D1 and D2 bounded by l, such that P = D1 ∪D2. Let D1 ⊂ X and A ⊂ D2, see Figure 2.18. Thus l is homotopically trivial and the secant is not strongly essential. By Schoenﬂies Theorem (Theorem 5.15), the separating sphere P in Figure 2.18 3 divides the ambient space S into two 3-balls. One arc of K lies in each of them, call them K1 and K2. As before, consider the loops α and β in the knot exterior, formed by the secant arc and an arc parallel to a portion of the knot. We have that α is a loop homotopic to a longitude of one of the two knots in which K is separated by P , and β is homotopic to a longitude of the other. By hypothesis, both of these knots are non-trivial. Therefore α and β are non-trivial in the fundamental group of the exterior of the respective component (a longitude of a non-trivial knot is always homotopically non-trivial). Then, by Lemma 2.37, α and β are also non-trivial in π1(X). Consequently, both the loops, α and β, are homotopically non-trivial and thus the secant is essential. Moreover, we can adopt another approach to show that actually this secant is not Secants Under Knot Composition 53 only essential but also non-peripheral. A composite knot is a knot that admits a swallow- follow torus in its exterior. This is a torus, like the one in Figure 2.19, which follows one component and swallows the other. It is a known fact that for composite knots these tori are incompressible and not boundary parallel [2, p. 86]. This means that their fundamental group injects into the knot group and since it is not boundary parallel this is a non-peripheral subgroup of π1(X).

Figure 2.19: Swallow-follow torus

Since α is a longitude of the swallow-follow torus, we have that the fundamental 2 group of such a torus is hµ, αi ' Z . On the other hand, the fundamental group of the 2 boundary torus is hµ, λi ' Z . Therefore, we have that α cannot be the power of a longitude, otherwise we would have: hµ, αi < hµ, λi. Therefore the secant is also non-peripheral.

Corollary 2.41. A nugatory crossing of a knot diagram D is not strongly essential. It is non-peripheral if and only if it is a nugaroty crossing of type II; in this case it is not simply inessential.

Proof. If a nugatory crossing is of type I, then the crossing is simply inessential (Lemma 2.29) and thus is neither essential nor strongly essential. If the crossing is nugatory of type II, then by the previous theorem it is non-peripheral but not strongly essential.

Note that we have already seen an example of a secant being essential but not strongly essential: the one corresponding to the arc in Figure 2.4. In particular, this secant is essential but peripheral, and since peripheral implies not strongly essential (Proposition 54 Essential and Strongly Essential Secants

2.23), this secant is also not strongly essential. Here instead we have seen examples of non-peripheral secants that are not strongly essential.

2.4 Smoothings of a Crossing

In this section we consider the essentiality of a crossing in relation to what happens if we desingularize it in order to obtain a two-component link, see Figure 2.20.

Figure 2.20: Desingularization of a crossing

Deﬁnition 2.42. Let D be a knot diagram and c one of its crossings. We denote by

Dc the diagram obtained from D by smoothing the crossing c in the way such that the resulting diagram represents a two-component link (that is, we smooth according to the orientation), see Figure 2.20.

Lemma 2.43. If Dc is the diagram of a split link then c is not a strongly essential crossing.

Proof. If Dc is a split diagram of a split link, it follows we claim that the crossing is nugatory. In fact, since after the smoothing the diagram has two connected components, we can isolate them with a disk. So, before the smoothing there will exist a circle intersecting the diagram only at the crossing, and thus the crossing is nugatory. But then, by Lemma 2.41, c cannot be strongly essential.

Let us now consider the case where Dc is a non-split diagram of a split link L. Since L is split, we can identify an embedded ball B in space such that one component lays inside it and the other does not intersect it. For orientations reasons, we can choose the loop lc to be on the boundary of B. In fact, when we smooth a crossing to obtain a two-component link, we have to smooth it according to the orientation, as in Figure 2.21. Smoothings of a Crossing 55

Figure 2.21: A crossing which give rise to a split link

Now, since B is embedded, we can slide lc along its boundary and isotope it to the trivial loop.

Remark 2.44. The inverse implication does not hold: There exists diagrams in which there is a not strongly essential crossings c and Dc is the diagram of a non-split link. For example if we switch any crossing of the trefoil, we obtain the unknot. In particular, we get a diagram where all the crossings are not strongly essential. Switching the marked crossing in Figure 2.20 before the smoothing, will give rise to the same smoothing, i.e. to a non-split link, but the original knot would be in this case the unknot.

Proposition 2.45. The diagram Dc is the diagram of a split link which has at least one unknotted component if and only if the crossing c is simply inessential.

Proof. As before, we have to distinguish the case in which Dc is a split diagram or not.

If Dc is split, then the crossing c is nugatory. But since one component of Dc is unknotted, the crossing must be nugatory of type I. Then, the result follows from Lemma 2.29.

If Dc is non-split if we consider the corresponding link L, there is an embedded disk bounded by one of its components. But this means that the loop α or β in the exterior of the knot represented by the diagram D is homotopically trivial.

To sum up, we have the following.

Proposition 2.46. Consider smoothing a crossing c in a diagram of a knot K in order to obtain a 2-component split link. According to the resulting link L we can classify the essentiality of it: 56 Essential and Strongly Essential Secants

1. L is the unlink link (i.e. a split link made of two trivial components), then c is simply inessential and K is the trivial knot.

2. L is split and one component is trivial, then c is simply inessential.

3. L is split and both component are non-trivial, then c is essential but not strongly essential.

Proof. For case 1, we have that by Lemma 2.43 c is not strongly essential. Moreover, since one component is trivial, the crossing is nugatory of type I and by Lemma 2.29, the crossing is simply inessential. But both the components are trivial, so the crossing is inessential from both sides (both α and β are trivial) and thus by Lemma 2.15, K is the unknot. Case 2 is a consequence of Lemma 2.43. Case 3 follows from Corollary 2.41.

If instead of considering the link we obtain desingularizing a crossing, we consider the knot we obtain switching a crossing, we can observe the following:

Proposition 2.47. Let K be a knot and D one of its diagrams. Let D0 be the diagram obtained from D by switching the crossing c to c0, and let K0 be the corresponding knot. 0 0 0 Call X the exterior of K and X the exterior of K . Then, if π1(X) 6' π1(X ), at least one crossing between c and c0 is strongly essential.

Proof. Consider the Wirtinger presentation obtained from D and D0. They coincide except for the crossings c and c0. Let us consider these crossings and add in both presentations an extra element and an extra relation, as in Figure 2.22.

Figure 2.22: Local diﬀerence between D and D0

We consider in both cases to use the generators g3 for the rest of the strands in the Smoothings of a Crossing 57

diagrams that go up from the crossings, until they meet an under-crossings, and g4 for all the rest of the strands going down from the crossings. We have the following presentations:

−1 −1 π1(X) = hGens, g1, g2, g3, g4 | Rels, g4g1 = g2 g3, g3 = g4i

0 −1 −1 π1(X ) = hGens, g1, g2, g3, g4 | Rels, g4g1 = g2 g3, g1 = g2i where Gens and Rels are all the relations and generators not occurring in the considered crossings. They are the same in the two presentations. Now, by assumption we ∼ 0 have: π1(K) =6 π1(K ). −1 Suppose that c is not strongly essential, then by Lemma 2.30 we have: g4g1 = −1 g2 g3 = id. So, the presentation of π1(X) is:

−1 −1 π1(X) = hGens, g1, g2, g3, g4 | Rels, g4g1 = g2 g3 = id, g3 = g4i = −1 −1 = hGens, g1, g2, g3, g4 | Rels, g3g1 = g2 g3 = id, g3 = g4i =

= hGens, g1, g2, g3, g4 | Rels, g1 = g2 = g3 = g4i

Now, suppose by contradiction that also c0 is non strongly essential. In this case, for similar reasons we get:

0 π1(X ) = hGens, g1, g2, g3, g4 | Rels, g3 = g4 = g1 = g2i

It is then easy to see that the two groups must be isomorphic. Speciﬁcally, the is the 0 map which sends the generator gi of π1(K) to the generator gi of π1(K ). Thus, c or c0 must be strongly essential.

Corollary 2.48. Let D be a knot diagram and c one of its crossings. If switching c leads to the unknot, then c is strongly essential.

Proof. This is a consequence of the previous result, of the fact that the only knot with knot group isomorphic to Z is the unknot and of the fact that all crossings in a diagram of the unknot are inessential.

For example, all crossings of a minimal diagram of the trefoil knot are strongly essential, as are the crossings responsible for the unknotting number one of the twist knots, see Figure 2.23. 58 Essential and Strongly Essential Secants

Figure 2.23: A twist knot Chapter 3

Secants of Alternating Knots

As we mentioned in the first chapter, alternating knots are those which admit an alternating diagram: A diagram in which over- and under-crossings appear in an alternating fashion when one follows it in one direction. The main advantage of dealing with only alternating knots is that it is possible to check many of their properties directly from their alternating diagrams. For example, a reduced alternating diagram of an alternating knot is minimal; this implies that the crossing number of the knot is the number of crossings of one such diagram. Moreover, all minimal diagrams of the same alternating knot can be related by a finite sequence of diagrammatical moves called flypes. Peter Guthrie Tait, a Scottish mathematician who computed the first knot table, introduced this diagrammatic move. He conjectured many facts about knots, most of which have been proved to be true long after his death. The best-known conjecture by Tait is the Flype Conjecture, which was proved to be true in 1993 by Menasco and Thistlethwaite [38].

Theorem 3.1 (Menasco and Thistlethwaite). Let K be a prime alternating link and let

D1 and D2 be two reduced alternating diagrams of K. It is possible to transform D1 into

D2 via a sequence of ﬂypes.

A flype is the local modification in a knot diagram depicted in Figure 3.1 (where we flip vertically a selected portion of the diagram):

59 60 Secants of Alternating Knots

F F

Figure 3.1: A ﬂype

In this chapter we will prove our main theorem: In a minimal diagram of an alternating knot all crossings are strongly essential. As we have seen in Theorem 1.52, all knots fall into one of three categories: torus knots, satellite knots and hyperbolic knots (remember that according to our deﬁnition composite knots are satellite). Since among torus knots only the (2, q)-torus knots are alternating (Proposition 1.47) and prime satellite knots are not alternating (Proposition 1.50), we have that all prime alternating knots are either (2, q)-torus knots or hyperbolic knots (Proposition 1.53). To prove our result about all alternating knots, we ﬁrst analyse prime knots. Then, we will consider their compositions using the fact that an alternating knot is composite if and only if any minimal diagram of it is diagrammatically composite (Theorem 1.43).

3.1 Hyperbolic Knots

Hyperbolic knots are very common: Most of the prime knots with small crossing number are hyperbolic. In fact, of all prime knots with crossing number less than 17, only 32 of the more than 1.7 million are not hyperbolic, see [27].

3.1.1 Hyperbolic 3-manifolds

In order to study the exterior of hyperbolic knots, we will now introduce some basic notions and properties about hyperbolic 3-manifolds. See [6], [43] and [36] for good references. In the following, we assume that all hyperbolic manifolds are orientable.

Deﬁnition 3.2. A manifold M is a hyperbolic manifold if it has a Riemannian metric of constant negative curvature equal to −1. Each point of the interior of M has a ball 3 neighbourhood that is isometric to a ball in H , the hyperbolic 3-space. Hyperbolic Knots 61

3 There are many models of H available, all isometrically diﬀeomorphic to each other. 3 3 Fist of all, let us consider the disk model of H . Take the unit open ball in R , let v = (x, y, z): 3 B = {v ∈ R : ||v|| < 1} and the hyperbolic metric on it: (dx)2 + (dy)2 + (dz)2 (ds )2 = 4 B (1 − (x2 + y2 + z2))2

3 The open ball B naturally compactiﬁes to the closed ball D by adding a sphere at 3 2 inﬁnity: ∂D = S∞. The geodesics in this model are the diameters and arcs of circles perpendicular to the boundary sphere. Figure 3.2 shows the analogous situation in dimension two.

2 Figure 3.2: Geodesics in the disk model of H

3 Let us now consider the upper half space model of H , we have

3 U = {x ∈ R : z > 0} and the hyperbolic metric on it: (dx)2 + (dy)2 + (dz)2 (ds )2 = U z2

2 The boundary is: ∂U = (R × {0}) ∪ {∞}, where {∞} denotes the point at inﬁnity. This 2 3 boundary is homomorphic to a sphere. In fact R × {0} is a plane in R and if we add {∞} we get a sphere. The geodesics are vertical half circles with their centers in the plane z = 0 and vertical lines. Figure 3.3 shows the analogous situation in dimension two. 62 Secants of Alternating Knots

2 Figure 3.3: Geodesics in the upper half plane model of H

A useful way of presenting orientable closed hyperbolic 3-manifolds is as quotients 3 of their universal cover H by a discrete group of orientation-preserving isometries of 3 H . For the knot exterior X, which is a manifold with boundary, the universal cover is 3 H minus the lifts of the solid tori corresponding to the missing tubular neighbourhood + 3 of the knot (as we will later see in more details). The elements of Isom (H ) can be classiﬁed in terms of their ﬁxed points:

+ 3 Deﬁnition 3.3. Let γ ∈ Isom (H ), and γ 6= id, then:

3 3 • γ is parabolic if it has one ﬁxed point in ∂H and no ﬁxed points in H ,

3 3 • γ is loxodromic if it has two ﬁxed point in ∂H and no ﬁxed points in H ,

3 3 • γ is elliptic if it has two fixed points in ∂H and infinitely many fixed points in H .

In the case of an elliptic transformation, there are two fixed points at the boundary 3 at infinity and an entire fixed hyperbolic geodesic in H . This geodesic joins the two fixed points at infinity, it is the axis of the transformation. For loxodromic isometries, the geodesic connecting the two fixed points at the boundary is again the axis of the transformation, it is again preserved, but this time not point-wise. Examples of parabolic transformations with fixed point at {∞} in the upper half space model are horizontal translations. In Figure 3.4 and Figure 3.5 we can see how these different hyperbolic isometries act and their axis: Hyperbolic Knots 63

Figure 3.4: Loxodromic and elliptic isometries on the disk model

Figure 3.5: A parabolic isometry on the upper half space model

The following theorem gives us a classiﬁcation of all the orientation preserving isometries of the hyperbolic 3-space, for a proof see [6]:

+ 3 Theorem 3.4. Let γ ∈ Isom (H ), and γ 6= id, then γ is either parabolic, loxodromic or elliptic.

The following proposition deals with the elements of ﬁnite order, see [36] for a proof:

+ 3 Proposition 3.5. An element of Isom (H ) has ﬁnite order only if it is of elliptic type.

We are interested in a particular type of group action:

Deﬁnition 3.6. A group G acts properly discontinuously on a topological space X if for every point x ∈ X there is an open neighbourhood Ux such that for all g ∈ G, g 6= id we have Ux ∩ g(Ux) = ∅.

Remark 3.7. A properly discontinuous action is free. In fact from the previous definition we have that every element except for the identity is fixed-point free in X: For g ∈ G and x ∈ X, gx = x implies g = id, which is the definition of a free action. 64 Secants of Alternating Knots

+ 3 3 Deﬁnition 3.8. A Kleinian group Γ is a subgroup of Isom (H ) that acts on H properly discontinuously.

Since according to our deﬁnition, properly discontinuous implies free, we have the following proposition, see [36, p. 26]:

3 Proposition 3.9. Let Γ be a Kleinian group. Then, the space H /Γ is a manifold.

Deﬁnition 3.10. A group G is torsion free if it has no element of ﬁnite order.

3 Remark 3.11. Let M ' H /Γ be a complete hyperbolic 3-manifold without bound- 3 ary, then the universal cover of M is H and the covering transformations are given by 3 π1(M) = Γ. It acts as a discrete group of isometries that do not ﬁx any point of H . See [36] for more details.

3 Proposition 3.12. Let H /Γ be a complete hyperbolic 3-manifold. Then Γ is torsion- free.

Proof. By the previous remark, γ ∈ Γ doesn’t have any fixed-point. The only elements that have fixed points are the elliptic ones, and by Proposition 3.5 they are the only ones of finite order.

3 Given a torsion-free Kleinian group Γ, the quotient H /Γ is a manifold and inherits a complete hyperbolic metric. Moreover, every complete hyperbolic manifold without boundary (and thus the complement of a hyperbolic knot) can be realized in this way. See [36, p. 26] for a proof:

Theorem 3.13. Let M be a complete hyperbolic manifold without boundary, then there 3 is a torsion-free Kleinian group Γ such that M ' H /Γ. The group Γ is unique up to 3 conjugation by orientation-preserving isometries of H .

Remark 3.14. The group of orientation-preserving isometries of hyperbolic 3-space can be identiﬁed with PSL2(C), the projective special linear group of 2×2 complex matrices. + 3 In fact the action of any isometry γ ∈ Isom (H ) extends to the boundary sphere of 3 2 1 H . The isometries on this sphere S ' CP are Möbius transformations and determine completely γ, see [36] and [43]. Hyperbolic Knots 65

The group PSL2(C) is isomorphic to SL2(C)/{±I}, which is the special linear group of complex matrices modulo plus or minus the identity. And SL2(C)/{±I} is isomorphic 1 to the Möbius group, which is the group of automorphisms of CP . a b If A = c d ∈ SL2(C), then ad − bc = 1. The corresponding transformation always has some fixed point in C ∪ ∞, because the equation (az + c)/(bz + d) = z admits at least one solution. The classification given in Definition 3.3 could have been attained considering the + 3 traces of matrices in PSL2(C), via the mentioned isomorphism: Let γ ∈ Isom (H ), with γ 6= id, and let A be the corresponding matrix in PSL2(C), then we have:

• γ is parabolic if |tr(A)| = 2,

• γ is loxodromic if |tr(A)| > 2,

• γ is elliptic if |tr(A)| < 2.

The following theorem describes Kleinian groups in terms of PSL2(C), see [36, p. 29] for a proof:

Theorem 3.15. A Kleinian group Γ is a discrete subgroup of PSL2(C).

3.1.2 Exterior and complement spaces of hyperbolic knots

Recall that we have defined hyperbolic knots as knots whose complement is a complete 3 hyperbolic manifold (Definition 1.51). On the one hand, the complement S \ K is not compact, but still has finite volume because, as we shall see, its end is a cusp. On the other hand, the exterior of a hyperbolic knot K is a hyperbolic manifold which is compact and has one toroidal boundary component (in the case of links of n components it will have n boundary components), and thus it has finite volume. As we have seen in Proposition 1.37, topologically the knot complement is the interior of the knot exterior. However, geometrically the situation is more complicated. To build the knot exterior from the knot complement we need to “chop off” some neighbourhood of the missing knot, in so forming a compact manifold with a flat torus boundary. However, there is not a unique way to perform this cut. We must choose the size of this tubular neighbourhood; so, the knot exterior is unique only up to this choice. Figure 3.6 is a 66 Secants of Alternating Knots representation of the complement of a hyperbolic knot with the end formed by smaller and smaller tori, and the exterior, with a flat torus as its boundary.

(a) (b)

Figure 3.6: The complement (a) and the exterior (b) of a hyperbolic knot

Given the knot complement of a hyperbolic knot, the hyperbolic structure on it is unique. This means that studying its geometry is useful for studying hyperbolic knots, since the hyperbolic structure is a knot invariant. The uniqueness of the geometric structure follows from the Mostow rigidity theorem [42], which states that the geometry of a complete ﬁnite-volume hyperbolic manifold of dimension at least 3 is unique. Join- ing this result with that of Gordon and Luecke [25], which states that two knots with homeomorphic complements are equivalent, we obtain that the hyperbolic structure is a complete knot invariant.

Let us now introduce some terminology in order to understand the ends of hyperbolic knot complements. For more details, see [58, Ch. 1].

3 Deﬁnition 3.16. A horosphere centred at p ∈ ∂H is a connected complete surface orthogonal to all geodesics ending at p. Any horosphere is isometric to the Euclidean plane.

3 A horosphere divides H into two connected regions homeomorphic to open 3-balls; 3 3 if we consider their closures in H ∪ ∂H , then the one with only one boundary point on 3 ∂H (in particular in p) is the horoball centred at p. Hyperbolic Knots 67

(a) (b)

Figure 3.7: Horoballs in the disk model (a) and half plane model (b)

3 Let C = S \ K be the complement of a hyperbolic knot. As we have already 3 mentioned, the universal cover of C is H . The end of C is a neighborhood of the missing 2 + 2 2 knot: T × R , where T × {∞} corresponds to the knot and T × 0 to the boundary of the neighborhood. As we will later see, the lift of any proper curve in the end of C 3 connects p ∈ ∂H to itself (this means that the end of C is “shrinking”). The horospheres 3 2 centred at p ∈ ∂H project to tori, which we can take as T × {e} ⊂ C and which have ﬂat metric.

2 Theorem 3.17. With the above notation, for t large enough, a neighborhood T × [t, ∞) 3 of the missing knot lifts in the universal cover H to a set of disjoint horoballs. Such a neighborhood is called a cusp.

2 For a ﬁxed “height” e > t on the cusp, we have a torus around K: T × {e}, which 3 lifts to the horosphere z = c in the half space model of H , where c varies monotonically with e. Let us consider a horoball centred at inﬁnity to which the cusp lifts. All isometries 3 sending this horoball to itself are parabolic isometries of H . The subset of these isometries in Γ ' π1(X) are the Euclidean translations generated by two translations. These translations are the lifts of the two generators of the fundamental group of the considered torus. In general, it is well known that any free non-trivial homotopy class in a Riemannian manifold can be represented by a geodesic. Now, choosing two appropriate generators 68 Secants of Alternating Knots

2 for the fundamental group of the torus T × {e}, a meridian µ and a longitude λ, these two loops lift in the universal cover to Euclidean geodesics which tile the plane into parallelograms. 3 3 We consider now H , the universal covers of C. If we represent H with the disk model, we have that the missing knot K is on the sphere at inﬁnity and a tubular neighborhood of K lifts to disjoint horoballs. A cusp C lifts to a set {Hi} of disjoint horoballs, with i ∈ I for some index set I. In the following we will omit to specify this index set, since it is always the same set I. If instead we look at the universal cover of the knot exterior, which we denote X˜ we ˜ 3 have that X ' H \ ∪iHi, since X is by deﬁnition the knot complement minus a cusp.

The boundary of X˜ is the set of all horospheres {∂Hi}.

Deﬁnition 3.18. Let M be a manifold with non-empty boundary. A closed loop γ in M is peripheral if it is a simple closed curve that can be homotoped through M into a curve in ∂M. We will also call the element [γ] ∈ π1(X) peripheral.

Algebraically an element of π1(M) is peripheral if it is a conjugate of an element of

π1(∂M); equivalently if its free homotopy class is the one of a loop in the boundary.

See [28, p. 118] for a discussion about peripheral elements in relation to parabolic isometries:

Lemma 3.19. Let M be a manifold with non-empty boundary. A loop γ in M is peripheral if and only if g([γ]) is parabolic, where g is the isomorphism from π1(X) to + 3 Γ ⊂ Isom (H ).

Deﬁnition 3.20. Let M be a compact 3-manifold with non-empty boundary. A subgroup G < π1(M) of the fundamental group of M is called peripheral if there exists a toroidal component T of ∂(M) such that, given the inclusion i : T,→ M, G is conjugate to a subgroup of i∗(π1(T )) that is isomorphic to Z × Z.

Note that this deﬁnition for hyperbolic knots is a consistent with Deﬁnition 2.4.

Deﬁnition 3.21. The manifold M is algebraically atoroidal if all subgroups of π1(M) isomorphic to Z × Z are peripheral.

We refer to [6, p. 157] for a proof of the following important property: Hyperbolic Knots 69

Proposition 3.22. A compact hyperbolic 3-manifold M with non-empty boundary is algebraically atoroidal.

3.1.3 Surfaces

Looking at the kind of surfaces that can be embedded in the exterior and complement of a hyperbolic knot is very useful for our research. In fact we will exploit the fact that the secant arcs we are interested in lie on a particular surface embedded in a knot exterior. This section collects a series of classical results, mostly from [60]; and a more recent result by Adams [3], which is crucial for the following analysis of essential secants. We begin by deﬁning incompressible surfaces, see [4]:

Deﬁnition 3.23. Let S be a compact surface embedded in a 3-manifold M, let D be an embedded disk in M such that S ∩ D = ∂D. Then D is a compressing disk for S if the curve ∂D does not bound a disk in S. If such a disk exists, then the surface S is compressible, otherwise, if it is not home- 2 omorphic to S , it is incompressible.

The following is a well-known characterization of incompressible surfaces: Geometri- cal incompressibility is equivalent to algebraic incompressibility. See [4]:

Proposition 3.24. A surface S is incompressible in M if and only if it is not homeo- 2 morphic to S and π1(S) → π1(M) is an injection.

We say that a submanifold N with non-empty boundary is properly embedded in a manifold M with non-empty boundary if (N, ∂N) ⊂ (M, ∂M).

Deﬁnition 3.25. Let S be a compact surface with non-empty boundary, properly embedded in a 3-manifold M with non-empty boundary. Let D be an embedded disk in M with ∂D = γ ∪ δ, where γ and δ are two arcs such that S ∩ D = γ, and D ∩ ∂M = δ. If there is an arc a in ∂S with the same endpoints as γ and δ and such that a is homotopic with ﬁxed endpoint to γ in S and to δ in ∂M, then D is a boundary compressing disk for S. If such a disk exist then the surface S is boundary compressible, otherwise it is boundary incompressible. 70 Secants of Alternating Knots

Again we have an equivalent algebraic characterization, see [4]:

Proposition 3.26. A compact surface with non-empty boundary S, properly embed- 2 ded in M is boundary incompressible if and only if it is not homeomorphic to D and

π1(S, ∂S) → π1(M, ∂M) is an injection.

Let Γ be a torsion-free Kleinian group. We have already mentioned that the action 3 3 3 2 3 2 on H extends to an action on the closure of H : H ∪ S∞ = H ∪ (R × {0}) ∪ {∞}. 3 Let Γ(x) denote the orbit of a point x ∈ H , Γ(x) := {γ(x): γ ∈ Γ}; since the action is properly discontinous, the accumulation points are located only in the sphere at inﬁnity. The set of all these accumulation points is well-deﬁned: It does not depend on the choice of the point x. See [36, p. 41].

Deﬁnition 3.27. The set Λ(Γ) of all the accumulation points of Γ(x) is called the limit set of Γ.

Note that the set Λ(Γ) is closed.

Deﬁnition 3.28. A Kleinian group Γ is quasi-Fuchsian if Λ(Γ) is a simple closed curve 2 and the two components of S∞ \ Λ(Γ) are preserved by the action of Γ. If the curve is a geometric circle, then the group is F uchsian.

Deﬁnition 3.29. Let X be the knot exterior of a hyperbolic knot and S an incompressible and boundary incompressible surface with boundary (not necessarily non-empty) properly embedded in X that is not homotopic to a boundary component of X. We have f : S,→ X and the induced injective map at the level of the fundamental groups: f∗ : π1(S) ,→ π1(X). The surface S is quasi-Fuchsian if f∗(π1(S)) is a quasi-Fuchsian subgroup of Γ. 3 Equivalently, S is quasi-Fuchsian if it lifts in the universal cover H to a disjoint union of planes, each with a limit set that is a simple closed curve. If the limit set is an actual circle, and the planes are geodesic, the surface is Fuchsian or totally geodesic, in this case f∗(π1(S)) is a Fuchsian subgroup of Γ.

Deﬁnition 3.30. An incompressible, boundary incompressible surface S with non-empty boundary, properly embedded in the exterior of a hyperbolic knot, is accidental if there Hyperbolic Knots 71 exists γ, a non-peripheral element of S, which is peripheral in X. Such curves are called accidental parabolic.

It is known [58, Ch. 1] that closed incompressible surfaces embedded in the exterior (or complements) of an hyperbolic knot, can only be of these two diﬀerent and mutually exclusive types: quasi-Fuchsian or accidental. For surfaces with non-empty boundary we have an extra possibility:

Definition 3.31. An incompressible and boundary incompressible surface with non- empty boundary in the exterior of a hyperbolic knot is geometrically infinite if its limit 2 set is the entire sphere at infinity S∞.

Results of Thurston [60] and Bonahon [9] lead to the following classiﬁcation, see [38] and [12]:

Theorem 3.32. An incompressible, boundary incompressible surface S with non-empty boundary, properly embedded in a hyperbolic knot exterior, is one of these mutually exclusive types:

1. quasi-Fuchsian,

2. accidental,

3. geometrically inﬁnite.

Geometrically infinite surfaces rise as virtual fibers in a virtually fibered knot. We say, using the definition of [64], that a 3-manifold M is fibered if M = S ×I/(x, 0) ∼ (φ(x), 1), where S is a surface and φ : S → S an orientation preserving homeomorphism. A 3- manifold M is virtually fibered if there is a finite cover of M that is fibered. A knot K in is fibered, or virtually fibered, if its exterior is fibred, or virtually fibred. See [52, Ch. 10], [64] and [65] for more information about fibered and virtually fibered knots.

Remark 3.33. A surface with non-empty boundary properly embedded in a hyperbolic knot exterior that is geometrically infinite is always homotopic to a virtual fiber in a virtually fibered knot. That means that a finite cover of the knot exterior is fibered and the surface is covered by one fiber. 72 Secants of Alternating Knots

This classiﬁcation will be important for the later discussion, in fact we will see that if a secant arc lies on a surface embedded in the exterior of a hyperbolic knot, which is not accidental, then the arc must be strongly essential.

Checkerboard Surfaces

For the following analysis of secant arcs corresponding to crossings in a diagram of a hyperbolic knot, we construct from a knot diagram particular well-studied surfaces properly embedded in the knot exterior and containing these secant arcs: the checkerboard surfaces. We follow the construction of Bao in [5]. Let K be a knot, and D a diagram on the 2 2 2 sphere: S ' R ∪ ∞. The diagram divides the surface of projection into regions: S \ K that can be canonically colored in a checkerboard coloring, i.e. satisfying the following conditions:

1. Each region must be colored either black or white.

2 2 2. The region containing ∞ is colored white (remember that we took S as R ∪{∞}).

3. For every arc of the diagram, the neighbour regions must be of diﬀerent colors.

Figure 3.8: A knot diagram with a checkerboard coloring

Remark 3.34. Every knot diagram admits a checkerboard coloring. See [5].

Now we can construct the checkerboard surfaces of a knot K associated to one of its diagrams D. Hyperbolic Knots 73

First of all pick a color, for example black. The black checkerboard surface is a 3 surface in R with boundary the knot K that projects to the black regions of the colored diagram injectively except at crossings, where a straight arc will map to a point. Each black region of the diagram corresponds to a disk plus a straight arc in the surface. Topologically a checkerboard surface is a collection of disks with twisted bands connecting them. The disks correspond to the regions of one color (they can be obtained by smoothing all the crossings appropriately) and the bands to the crossings of the diagram (in order to form the bands, we take a part of the disks corresponding to the part of the regions close to the crossings). Using this method it is possible to obtain the two checkerboard surfaces, the black and the white one. In order to obtain a surface that is embedded in the knot exterior, we take the intersection of the chekerboard surfaces with X. Call these two surfaces CK1 and CK2. The boundaries of these surfaces are longitudes in the boundary torus of X.

The secant arcs corresponding to crossings of a knot diagram lie in both CK1 and

CK2. More speciﬁcally, these secant arcs are vertical arcs embedded in those surfaces and they project to a crossing of the diagram. The following is an important result that will be crucial for our following analysis of strongly essential secants, it has been proved by Adams [3].

Theorem 3.35 (Adams). Any checkerboard surface associated to any minimal diagram of an alternating hyperbolic knot is quasi-Fuchsian. In particular it does not contain any loop representing an accidental parabolic element.

Sketch of proof. A checkerboard surface associated to a reduced alternating diagram of hyperbolic knots is incompressible and boundary-incompressible [38, Proposition 2.3]. As we have seen in Proposition 3.32, an incompressible, boundary-incompressible surface in a hyperbolic knot complement can be of one of the following three types: quasi-Fuchisan, accidental or geometrically inﬁnite with limit set the entire boundary of 3 H . 3 Adams proved that for any reduced alternating diagram of a hyperbolic knot in S , the associated checkerboard surface is quasi-fuchsian [3, Theorem 1.9] and therefore does not contain any simple closed curve representing an accidental parabolic. 74 Secants of Alternating Knots

In the case that a checkerboard surface is orientable, it is isotopic by construction to the Seifert surface constructed using Seifert’s original algorithm [56]. It is a known result that a minimal genus Seifert Surface of a non-fibered knot is always quasi-Fuchsian [23]. In the case of alternating knots, applying Seifert’s algorithm to a minimal diagram gives rise to a minimal-genus Seifert surface [44]. Even if the knot is fibered, this Seifert surface is not a virtual fiber and thus not geometrically infinite [3]. Therefore this Seifert surface is quasi-Fuchsian and it is isotopic to a checkerboard surface, which is therefore also quasi-Fuchsian. Moreover Adams has proved that the two checkerboard surfaces must both be quasi-Fuchsian [3].

Thistlethwaite and Tsvietkova [59] deﬁne taut diagrams for hyperbolic knots as diagrams whose associated checkerboard surfaces are incompressible, boundary incompressible and do not contain any closed curve representing an accidental parabolic element (i.e. they are not accidental surfaces). With this new terminology, Proposition 3.35 implies that minimal diagrams of alternating hyperbolic knots are taut.

3.1.4 Secants

Let S denote a secant of a knot K and A a corresponding secant arc, which is a proper arc in the exterior of a knot K. Let α be the corresponding loop, as before. As we saw, we can represent the exterior space X as a quotient of its universal cover:

˜ 3 X ' X/Γ = (H \ ∪iHi)/Γ where {Hi} are the disjoint horoballs to which the interior of N(K) lifts and Γ ' π1(X) ⊂ + 3 Isom (H ) is formed by hyperbolic and parabolic isometries. That is, we have an isomorphism:

g : π1(X) → Γ

We consider A˜, one of the lifts of A in the universal cover. Note that an arc lifts to many arcs, we pick one by choosing one lift of its starting point.

Since ∂A ⊂ ∂X, we have ∂A˜ ⊂ ∂X˜ = ∪i∂Hi. The boundary points of A˜ are on the horospheres to which the boundary torus of X lifts, see Figure 3.9. The secant S lifts 3 3 instead to a path in H with endpoints in the the boundary of H , which are two ideal points. Hyperbolic Knots 75

(a) (b)

Figure 3.9: A path in X (a) and one of its lifts (b)

If the path is peripheral, then it is homotopic to an arc in the boundary torus and, as we shall see (Proposition 3.37), its lifts will start and end in the same horosphere, see Figure 3.10.

(a) (b)

Figure 3.10: A peripheral path in X (a) and one of its lifts (b)

Proposition 3.36. Let A be a secant arc. Then, A is non-peripheral if and only if [α] 3 is non-peripheral, or equivalently if g([α]) is a hyperbolic isometry of H . We can replace α by β.

Proof. By deﬁnition we have that A is peripheral if and only if α is free homotopic in X to a loop in the boundary torus, that is, if and only if [α] is non-peripheral. 76 Secants of Alternating Knots

Then, by Lemma 3.19, [α] is peripheral if and only if g([α]) is a parabolic element of + 3 Γ ⊂ Isom (H ). Since Γ does not contain any elliptic elements, [α] is non-peripheral if and only if g([α]) is hyperbolic. We can replace α by β using Remark 2.6.

Recall that the boundary torus of X lifts to the set of disjoint horospheres {∂Hi}.

Proposition 3.37. A secant arc A is non-peripheral if and only if its lifts in X˜ are arcs connecting two diﬀerent horospheres in {∂Hi}.

Proof. Pick one lift A˜ of A in X˜, as we have seen, this is an arc that connects two horospheres among ∪i∂Hi. The arc A is non-peripheral if and only if it is not homotopic to an arc in the boundary torus. If A is homotopic to an arc in the boundary torus, then, by the homotophy lifting property, A˜ will be homotopic to an arc in the lifts of this boundary torus: ∪i∂Hi. On the other hand, if the lift A˜ is homotopic to an arc in ∪i∂Hi, then, projecting down, A is homotopic to an arc in the boundary torus. So, A is non-peripheral if and only if the arc A˜ is homotopic to some arc in ∂X˜ =

∪i∂Hi. ˜ 3 But since X = H \ ∪iHi is simply connected and the horospheres ∂Hi are disjoint, an arc is homotopic to an arc in ∂X˜ = ∪i∂Hi if and only if has the endpoints in the same horosphere Hi. Then we get the result: A secant arc A is non-peripheral if and only if its lifts in X˜ are arcs whose endpoints belong to different horospheres in the set {∂Hi}. Note that the choice of the particular lift of A is not significant because all the different lifts are images of A˜ by some element of the deck transformation group.

Corollary 3.38. Let S be a secant. Then, S is non-peripheral if and only if any of its 3 3 lift in H is an arc homotopic to a geodesic in H . Proof. This is a consequence of the previous proposition. The secant S is a proper arc in the knot complement C with well deﬁned limit points for each end (this is because the end of the knot complement is a cusp). If we take a lift S˜ of S, its endpoints are in the sphere at inﬁnity. Hyperbolic Knots 77

These endpoints are distinct if and only if any of its lift A˜ is an arc connecting two 3 diﬀerent Hi in H . In fact they correspond to the centres of the horosphere to which the endpoints of A˜ belong. 3 But the endpoints of A˜ are distinct if and only if A˜ can be homotoped in H to a geodesic connecting them.

Figure 3.11: An arc homotopic to a geodesic connecting the centers of two horoballs in 3 H

Theorem 3.39. Let D be a minimal diagram of an alternating hyperbolic knot. Then, all crossings of D are non-peripheral.

Proof. Let D be a minimal diagram of an alternating hyperbolic knot and Σ one of its checkerboard surfaces. The secant arcs corresponding to the crossings of D lie in Σ, which is quasi-Fuchian (Theorem 3.35). Since we are only dealing with prime reduced diagrams, we claim that any secant arc A corresponding to a crossing of D is a non-separating arc in Σ. In other words, we claim that if such an arc was separating in a checkerboard surface, the corresponding crossing would be reducible (either nugatory or an extra-crossing in a composite knot). In fact, if such a proper arc is separating, then by cutting Σ along it we obtain a surface with two connected components. In particular, also the boundary of Σ, which is a longitude in the boundary torus of X, is divided into two disjoint components. Projecting the boundary of the surface, we get an equivalent knot diagram (note that the boundary of the checkerboard surfaces are curves in the boundary torus of the knot exterior isotopic to the knot). 78 Secants of Alternating Knots

Then A projects to a crossing and if it is separating, so there is a circle intersecting the diagram in only a crossing, and therefore the crossing is a nugatory crossing. So, all the secant arcs A relative to crossings are non-separating proper arcs in Σ. But then the non-separating secant arc A is in particular non-peripheral in Σ. In fact, if the arc was peripheral, it would be homotopic in Σ to a curve in the boundary of Σ (which is a longitude in the boundary torus of X) and thus cutting along it it would divide the surface in two connected components. The surface Σ is quasi-Fuchisan and thus not accidental, therefore A is also non- peripheral in X, that is, it is not homotopic to a curve in the boundary of X.

By Proposition 2.23 if an arc is peripheral, then it cannot be strongly essential. In the case of hyperbolic knots the converse also holds.

Proposition 3.40. If a secant arc A of a hyperbolic knot is non-peripheral, then it is strongly essential.

Proof. We claim that if A is not strongly essential, then it must be peripheral. Consider the elements α and µ previously introduced. Since they are elements of

π1(X), they correspond to isometries of hyperbolic space. The element µ is the homotopy 3 class of a meridian curve and therefore it corresponds to a parabolic isometry of H . Since α is in the fundamental group of X, it cannot be of elliptic type; in particular it has infinite order. Then we consider their commutator, since A is not strongly essential, we have: [α, µ] = 0. But then the group hα, µi generated by α and µ is abelian. We have that α 6= µ since they have different linking numbers with the knot. Moreover, all the elements of the group they generate will have infinite order (since π1(X) does not contain any element of finite order) and thus they must generate a group isomorphic to Z × Z. But by Proposition 3.22, X is algebraically atoroidal, thus this group is conjugate to the subgroup of π1(X) corresponding to the boundary torus. This implies that both µ and α are peripheral elements, because their images in + 3 Isom (H ) are parabolic elements (Lemma 3.19). Both these elements share the same (2, q)-Torus Knots 79

fixed point in the sphere at infinity and they fix any horosphere centred in the fixed point. So, by Proposition 3.36, A is peripheral.

Therefore, in the case of hyperbolic knots, the notion of strongly essential and non- peripheral coincide:

strongly essential ⇐⇒ non-peripheral

Using this fact, we prove that in a minimal diagram of an alternating hyperbolic knot all crossings are strongly essential, and in particular also essential.

Theorem 3.41. Let D be a minimal diagram of an alternating hyperbolic knot. Then all secants corresponding to crossings of D are strongly essential.

Proof. In Theorem 3.39 we proved that all the crossings of a minimal diagram of an alternating hyperbolic knot are non-peripheral. In the previous proposition we showed that if a crossing of a hyperbolic knot is non-peripheral, then it is strongly essential, thus the claim.

3.2 (2, q)-Torus Knots

All torus knots are prime but only (2, q)-torus knots are alternating (Proposition 1.47). In this section we will prove that all crossings in a minimal diagram of a (2, q)-torus knot are strongly essential.

Figure 3.12: Standard diagram of a (2, q)-torus knot

First of all we need the following lemma, which is a known application of the ﬂype conjecture: 80 Secants of Alternating Knots

Lemma 3.42. A minimal diagram of a (2, q)-torus knot is equivalent to the standard 2-strand braid diagram (see Figure 3.12).

Proof. The standard braid diagram of a (2, q)-torus knot (see Figure 3.12) is alternating and reduced, thus it is minimal. By Theorem 3.1, all minimal diagrams of an alternating knot are related by flypes. Then, from this diagram we can obtain all of the other minimal diagrams of the same knot via flypes. In order to perform a flype we have to isolate a “box” to be flypes and a crossing 2 outside it in a knot diagram D. This box is homeomorphic to a disk D , such that 2 D ∩ ∂D is made of four points. We claim that the only possible boxes in the standard braid diagram of a (2, q)-torus knot are the ones containing adjacent crossings. The box cannot contain any crossing neither a portion of the upper strands, otherwise its endpoints would be six or eight instead of four, Figure 3.13.

Figure 3.13: Flypes in a braid diagram of a (2, q)-torus knot

For the same reason we cannot put non-adjacent crossings in the box. If we want to have a box instead of a strange shaped disk we can simply isotope the diagram as in Figure 3.14.

Figure 3.14: Flypes in a braid diagram of a (2, q)-torus knot

Thus, the box must contain a series of adjacent crossings. Applying a flype to such (2, q)-Torus Knots 81 a box and an external crossing leaves the structure of the diagram unaltered. In fact it only “translates the crossings”, see Figure 3.15 (for convention we picked the crossing outside the box to be on the left of the box, but the same would apply if we took the crossing on the right of the box instead). Note that in the figure we can put an arbitrary number of crossings inside the box of the flype.

Figure 3.15: Flypes in a braid diagram of a (2, q)-torus knot

Theorem 3.43. All crossings in a minimal diagram of a (2, q)-torus knot are strongly essential.

Proof. Using Lemma 3.42, we only need to analyse the standard diagram of a (2, q)-torus knots. The loops we have to consider in order to check the strong essentiality of a crossing are the ones with linking number zero with the knot. Figure 3.17 depicts one of them.

Let us look at the Wirtinger presentation for π1(X). We have q generators and q relations, with q odd:

π1(X) = hg1, . . . , gq|g1g2 = g2g3 = ··· = gqg1i as you can see from Figure 3.16 for q = 5. 82 Secants of Alternating Knots

g g5 2 g3 g4

Figure 3.16: Wirtinger presentation of π1(X)

By the combinatorial structure of this presentation, cyclic permutations of the generators do not change the relations. Call φ : {g1, . . . , gq} → {g1, . . . , gq}, such that gi 7→ gi+1. Then, φ induces an automorphism of π(X). −1 −1 −1 Now, the loops we are interested in are of the form: g1 g2, g2 g3, . . . gq g1. We have: −1 −1 φ(gi gi+1) = gi+1gi+2. So, they are all images of each other by an automorphism of

π1(X). In particular, if one is non-trivial, then all are non-trivial.

Figure 3.17: A loop relative to a crossing

By Theorem 2.31 we know that at least one crossing for every diagram of any non- trivial knot is strongly essential, thus all of them have to be so. Therefore all crossings of a minimal diagram of a (2, q)-torus knot are strongly essential.

3.3 Composite Knots

Every minimal diagram of an alternating composite knot is diagrammatically composite (Theorem 1.43). Using Shubert’s decomposition theorem (Theorem 1.44), we can decompose any alternating knot in prime alternating factors. Since a minimal diagram is always alternating and reduced, all the crossings correspond to the ones of the factors. In fact we have additivity of crossing number for alternating knots [2]. Composite Knots 83

Lemma 3.44. In any minimal prime diagram of an alternating knot all crossings are strongly essential.

Proof. As we have seen, all prime alternating knots fall into the category of hyperbolic knots and (2, p)-torus knots. The result holds for the ﬁrst category (Theorem 3.41), as well as for the second (Theorem 3.43).

Theorem 3.45. In any minimal diagram of an alternating knot all crossings are strongly essential.

Proof. By the previous lemma, the result holds for all prime alternating knots. Let then D be a minimal diagram of a composite alternating knot. By Theorem 1.43 D is diagrammaticaly composite. Therefore by Proposition 1.45 we can decompose D into prime alternating diagrams. By Theorem 2.38, when we compose two knot diagrams the strongly essential crossings are still strongly essential crossings as crossings of the composed diagram. Therefore we have the result. 84 Secants of Alternating Knots Chapter 4 n-Essential Secants

In this chapter we introduce a new and stronger characterization of secants and secant arcs of knots: n-essentiality. We consider the homotopy class of a lift of the loop lA— which we investigated in order to deﬁne the strong essentiality of secant arc—to the n- 3 fold cyclic branched covers of S branched along the knot. The loop lA has by deﬁnition linking number zero with the knot. Thus, it lifts as closed loops to any cyclic cover of the 3 knot exterior and consequently also to any cyclic branched covers of S along the knots. If the free homotopy class of one of these lifts is non-trivial in the fundamental group of the n-fold branched cover, then we say that the corresponding secant arc is n-essential. Moreover in this chapter, we will analyze 2-essentiality in detail. Doing so, we use results about the fundamental group of the double branched cover to prove existence statements for 2-essential crossings of knot diagrams.

4.1 Cyclic Branched Covers of Knots

Let us start by deﬁning covering spaces:

Deﬁnition 4.1. Let M and N be topological spaces. A continuous surjective map p : M → N is called a n-sheet cover of N if for all x ∈ N there exists an open neighbourhood B of x, such that p−1(B) ' B × T , where T is a discrete space with n points, i.e. p−1(B) is homeomorphic to the disjoint union of n copies of B. Moreover, the following diagram must be commutative:

85 86 n-Essential Secants

' p−1(B) - B × T @ p @ q @R © B where q is the standard projection of B × T into B. A deck transformation (or covering transformation) is a map d : M → M such that the following diagram is commutative: d MM- @ p @R © p N d is an automorphism of the space M that preserves the cover.

In particular, d permutes the sheets of the cover. Assume the base space N is connected. We say that a n-fold cover has n sheets, which are permuted by the deck transformations. In general, it is impossible to separate these sheets. We can deﬁne the sheets using an evenly covered open neighbourhood B, whose pre-image is a collection of n −1 open neighbourhoods Bi, i = 1, . . . n homeomorphic to B such that π |Bi : Bi → B are homeomorphisms. Each of these Bi is a sheet over B. Given a topological space X, by Hurewicz’ Theorem (Theorem 5.17) the abelianization of the fundamental group π1(X) is the ﬁrst homology group H1(X). For knot exteriors we have the following standard result:

Theorem 4.2. Let K be a knot and X its exterior. There is an isomorphism H1(X) → Z such that α 7→ lk(K, α).

Proof. The ﬁrst homology group of a knot exterior is the abelianization of the knot group, thus it is isomorphic to Z and it is generated by a meridian loop in the boundary torus. A meridian is a loop in ∂N(K) that bounds a disk D in N(K) such that D ∩ K is one single point. Then, the homology class of each loop in X is determined exclusively by its linking number with the knot. For a detailed proof, which makes use of the Mayer- Vietoris sequences for homology, see [34, p. 12].

Using the well-known correspondence for connected and locally path-connected spaces between the subgroups of the fundamental group of a manifold and its connected covering spaces, we can deﬁne the cyclic covering spaces of X as follows. Cyclic Branched Covers of Knots 87

For every subgroup H < π1(X) there exists a connected covering:

pH : YH → X with (pH )∗π1(YH ) = H, where the map (pH )∗ is the induced map from pH at the level of fundamental groups. To construct such a correspondence we can start from the universal cover of X, which by deﬁnition has trivial fundamental group, see [26].

◦ 3 ˜ Deﬁnition 4.3. The n-fold cyclic cover of X ' S \ N(K), which we will denote by Xn, is the covering 3-manifold whose fundamental group is the kernel of the composition of the following two maps:

abelianization projection π1(X) −−−−−−−−−→ Z −−−−−−→ Zn

Call this map φn:

φn : π1(X) → Zn

˜ The subgroup π1(Xn) of π1(X) is the subgroup of all loops with linking number nZ with the knot. Consider the Wirtinger presentation for π1(X). The positive Wirtinger generators have linking number +1, therefore we will have to combine n of them in order to obtain a loop that lifts as a closed loop to the n-fold cyclic cover. For example, π1(Xñ) contains all words of length a multiple of n composed by positive Wirtinger generators. We will use the cyclic covers of a knot exterior in order to construct the cyclic branched 3 covers of S over that knot. But first we have to define branched covers in general. Basically these are regular covers everywhere, except for the branching set that is only covered once:

Deﬁnition 4.4. Let M and N be manifolds and let L be a submanifold of N. Then π : M → N is a n-sheet branched cover of N with branching set L if:

1. the restriction of π to M \π−1(L) is a n-sheet cover of N \L, which is the associated unbranched cover,

2. the restriction of π to π−1(L) is a homeomorphism onto L. 88 n-Essential Secants

In the following lemma we present three important and well-known examples of branched covers that we will later need.

Lemma 4.5. For any n ∈ N we have:

2 2 1. The n-fold cyclic branched cover of D branched over one point is D itself.

2 2 2. The n-fold cyclic branched cover of S branched over two points is S itself.

3 1 3 3. The n-fold cyclic branched cover of D branched along an unknotted arc D is D itself.

3 1 3 4. The n-fold cyclic branched cover of S branched along an unknotted circle S is S itself.

2 2 2 Proof. For the ﬁst case consider D as the unit complex disk. Then the map D → D , n such that z 7→ z is a covering map: By this map, the unit disk in C is covered by itself n times except for the origin that is covered just once. 2 In fact, we can observe that the quotient of a disk D by an n-fold rotation around its center is again a disk. This rotation is the deck transformation of the covering map 2 and D is evenly covered n times, except for the branching point, see Figure 4.1.

2 Figure 4.1: D as a branched cover of itself over a point

2 2 Given D as a branched cover of itself over a point, we can construct S as a branched cover of itself over two points simply by gluing two 2-disks along their boundaries. In this way we get again a n-fold rotation as covering transformation. This time the ﬁxed points are two: the centers of the two disks. Cyclic Branched Covers of Knots 89

2 Figure 4.2: S as a branched cover of itself over two points

3 2 1 The 3-ball D is homeomorphic to a solid cylinder D × D and in this form we can apply the same rotation again. In this case the ﬁxed-point set is the straight segment 2 corresponding to the centers of all the stacked D s. 3 For the last case we construct S by gluing two 3-balls along their boundary. If we keep the rotation we have a n-fold cover, this time it is branched along the union of two unknotted arcs, that is along an unknotted circle.

3 In particular, we are interested in branched covers of S with a knot as branching set:

3 Deﬁnition 4.6. The n-fold cyclic branched cover of S branched along K is the branched cover whose associated unbranched cover is the n-fold cyclic cover of the knot complement and whose branched set is the knot K. For simplicity we will call it the n-fold cyclic branched cover of K and denote it by Σn(K). This space is obtained from the unbranched cover by adding the missing knot to the knot complement: 3 π : S^3 \ K ∪ K → S

Remark 4.7. The n-fold cyclic branched cover of K is a closed 3-manifold which can be obtained by gluing a solid torus to X˜n.

The boundary torus of X is covered by the boundary torus of Xñ, the n-fold cyclic cover of the knot exterior. The manifold Σn(K) is obtained by gluing a solid torus 2 1 ˜ T' D × S to the boundary of Xn such that the meridian of T is identified with the lift of n-times the meridian of ∂X to ∂Xñ. See [11, p. 117]. 90 n-Essential Secants

A way to construct cyclic branched covers of knots is through Seifert surfaces (these are orientable surfaces with the knot as boundary), as explained by Rolfsen [52]. Consider 3 a knot K and a Seifert surface S(K) for it. Then cut S along S(K), and call the resulting manifold N. Its boundary is made of two copies of S(K): S(K)+ and S(K)−, joined along their common boundary K. Then take n copies of N and glue them together along their boundary in such a way that S(K)+ in the i-th copy is identiﬁed with S(K)− in the (i + 1)-st copy, where the index i is considered modulo n. The resulting manifold is the n-fold cyclic branched cover of K. There is an action of Zn by deck transformations on the resulting manifold. This action permutes the n copies of N while the knot, which is the common boundary of S(K)+ and S(K)− remains ﬁxed. See [1] and [48] for more information on cyclic branched covers.

Deﬁnition 4.8. Let K be an oriented knot. The inverse −K of K is obtained by inverting the orientation of K.

The invertibility (i.e. the property of being equivalent to its inverse) is a knot invariant.

3 Remark 4.9. The n-fold cyclic branched cover of S over a knot does not detect its orientation:

Σn(K) ' Σn(−K)

This can be derived from the fact that in the deﬁnitions orientations were not mentioned.

The cyclic branched cover can be used to recognize the unknot, this is a known result:

Theorem 4.10. Let K be a knot and Σn(K) its n-fold cyclic branched cover. If K is 3 3 trivial, then Σn(K) ' S for all n. If K is non-trivial, then Σn(K) 6= S for all n.

Proof. If K is trivial, then the result follows from Lemma 4.5. The other implication is a consequence of the Smith Conjecture (Theorem 5.19). In 3 fact if we have a n-fold branched cover of S over a knot, associated to it there is a deck transformation of order n with ﬁxed-point set exactly the knot K (this transformation permutes the sheets of the cover leaving the branching set ﬁxed). Cyclic Branched Covers of Knots 91

3 3 If the branched cover is S , this transformation is a homeomorphism of order n of S to itself with the knot as ﬁxed-points set. Then, by the Smith Conjecture, the knot must be the unknot.

Next we consider what happens to the branched cover when we compose two knots. First we need the following lemma:

Lemma 4.11. Let K be a knot and S a sphere intersecting it in two points such that it 3 divides S in two 3-balls B1 and B2, and such that B2 ∩ K is an unknotted arc. Then, the n-fold cyclic branched cover of B1 branched along B1 ∩ K is homeomorphic to the branched cover over the knot, minus a ball containing a trivial arc.

Proof. The sphere S intersects the knot in two points. The ball B2 contains an unknotted arc. Then, by Lemma 4.5, the branched cover of this ball is again a 3-ball. This implies that B1 must be sent by the cover to the branched cover of the original knot minus a ball with an unknotted arc inside.

Proposition 4.12. Let K = K1#K2 be a composite knot. For every n ∈ N we have:

Σn(K) ' Σn(K1)#Σn(K2)

Proof. If one of the summands is the unknot, then we can apply Theorem 4.10 and use the fact that the 3-sphere is the identity element for the operation of connected sum on 3-manifolds. The case n = 1 is trivial since the 1-fold branched cover of any knot is the 3-sphere. For n > 1 and for non trivial knot compositions, we use Lemma 4.5. By Deﬁnition 2 3 1.41 there exists a separating sphere S which divides S in two solid balls, both of which contain a non-trivial portion of the composite knot. This sphere will meet the knot in exactly two points, which will be part of the branching set. Since any branched cover of a sphere over two points is again a sphere, this sphere is lifted by the cover to another sphere, which is also intersecting the knot in two points.

We only have to check that this new sphere in the branched cover bounds Σn(K1) and Σn(K2). To do so we follow what happens to the ball-arc pairs in the branched 92 n-Essential Secants cover. By Lemma 4.11, these are sent to the branched cover of one component of the knot minus a ball with a trivial arc. Then, these ball-arc pairs are sent exactly to Σn(K1) and Σn(K2) minus a ball with a trivial arc. So, the space Σn(K) is the connected sum of Σn(K1) and Σn(K2).

Even if, as we have seen, it is possible to distinguish the unknot via branched covers, it is not possible to distinguish all knots in this way. In fact for all n > 1 there exists non-equivalent knots with homeomorphic n-fold cyclic branched covers. For example this is the case for some composite knots [61,62]. Let K be an oriented knot that is not invertible. Then K#K and K# − K are non-equivalent knots but have homeomorphic cyclic branched covers. In fact by Remark 4.9, the orientation of the knot is not detected by the cover. Examples of non-equivalent knots with homeomorphic double branched covers are many: For instance all Conway mutants (a mutant knot can be obtained from a knot diagram by isolating a tangle and substituting it by a reﬂection of itself) have this property [13,31,39].

4.2 n-Essentiality

Using the notation already introduced in the previous chapter, let K be a knot and D one of its diagrams. Let c be crossing of D and lc be the loop around it that we analyzed to check the strong essentiality of the crossing.

Deﬁnition 4.13. Let K be a knot. A secant arc A of K, and the corresponding secant

S, are n-essential if the free homotopic class of a lift of lA to Σn(K) is non-trivial.

Remark 4.14. If one of the lifts of lA is freely homotopically trivial, then so are all the others. That is why n-essentiality is well-deﬁned.

The same deﬁnition works for crossings as well. Like before, we consider the secant connecting the two strands meeting in a crossing.

Remark 4.15. All secants are 1-inessential. In fact, this is a degenerate case because the 3 1-fold branched cover of any knot is S itself. n-Essentiality 93

Firs we clarify the hierarchy between these diﬀerent new kinds of essentialities. As a preliminary task, we will to express the fundamental group of the n-fold cyclic branched covers from a presentation of the fundamental group of the cyclic (unbranched) n-fold cover. To do so, we use a well-known application of the theorem of Van Kampen (Theo- rem 5.18):

Proposition 4.16. The fundamental group of the n-fold cyclic branched cover over a knot K is isomorphic to the quotient of the fundamental group of the n-fold cyclic cover of the exterior of K modulo the normal subgroup generated by the n-th power of a meridian loop:

n π1(Σn(K)) ' π1(X˜n)/hhµ ii

◦ 3 Proof. Let K be a knot and X its exterior. Then X = S \ N(K).

When we add the solid torus N(K) to the cover we get: X˜n ∪ N(K) ' Σn(K) the cyclic branched cover over K (Remark 4.7). ˜ 2 The intersection Xn ∩ N(K) ' T is a torus. In order to apply the theorem of Van Kampen (Theorem 5.18) to these spaces we have ◦ ◦ ◦ ˜ 0 1 2 to choose appropriate open subsets of Σ2(K). Let U :=Xn and V :=N (K)' S × D , ◦ where N(K) ⊂N 0(K). We glue them so that they overlap, along a neighbourhood of the ◦ 2 boundary components of their closures: T × I. We then get the following commutative diagram:

i j 1: U XXXz1 ◦ 2 Σ (K) ' U ∪ V U ∩ V ' T × I 2 XXz : i2 V j2

We have that the fundamental group of the interior of the knot exterior, is isomorphic to ◦ 2 2 the fundamental group to the knot exterior. Moreover π1(T × I) = π1(T ). Thus, the maps induced onto the fundamental groups are the following:

˜ i π1(Xn) j1∗ 1∗ : XXz 2 X π1(Σn(K)) Z XXX : i2 Xz ∗ Z j2∗ 94 n-Essential Secants

2 The fundamental group of the torus is Z . It is generated by a meridian m and a longitude l. Then we have: ˜ π1(Xn) ∗ Z π1(Σn(K)) ' −1 −1 hhi1∗ (m)(i2∗ (m)) , i1∗ (l)(i2∗ (l)) ii ◦ The image of the meridian loop in the fundamental group of the solid torus N 0(K), ˜ n i2∗ (m) is trivial, and the image of it in Xn is µ . Therefore we have:

−1 −1 n −1 hhi1∗ (m)(i2∗ (m)) , i1∗ (l)(i2∗ (l)) ii = hhm , i1∗ (l)(i2∗ (l)) ii ◦ 0 The group Z ' π1(N (K)) is generated by i2∗ (l) and by the second relation given by the amalgamation, this generator is identiﬁed with the loop i1∗ (l). Thus we have: ˜ ˜ π1(Xn) ∗ Z π1(Xn) π1(Σn(K)) ' n −1 ' n hhm , i1∗ (l)(i2∗ (l)) ii hhµ ii as wanted.

Proposition 4.17. For any n ∈ N, if a secant is n-essential, then it is strongly essential.

Proof. First note that by Remark 4.15 the case n = 1 never occurs.

Let A be a secant arc. If this is not strongly essential, then the loop lA around it is trivial in π1(X). But any lift of a trivial loop is also trivial in any cyclic branched cover, and thus the secant cannot be n-essential for any n ∈ N.

Therefore if a secant is n-essential it must be strongly essential, however the reverse statement does not hold. We now show that these new notions are striktly stronger than the ones introduced before.

Proposition 4.18. For all n ∈ N, there are secants that are strongly essential but not n-essential.

Proof. Consider the trefoil knot: T (2, 3). The Wirtinger presentation, obtained from a minimal diagram (like the one in Figure 4.3(a)) of it, is the following:

G = hx, y, z|xy = yz, yz = zx, zx = zyi = hx, y|xyx = yxyi n-Essentiality 95 we can reduce the generator set to two elements since we have z = xyx−1.

Now we consider a loop around a secant, which always has the form of a commutator [µ, g], where µ is a meridian loop. Note that as meridian we can take any Wirtinger generator.

(a) (b)

Figure 4.3: Minimal diagrams of the trefoil knot

We claim that the loop: [x, yn] = xynx−1y−n is the loop around a secant that is strongly essential but not n-essential. First we will find the corresponding secant. Con- sider the case n = 2, we can see in Figure 4.4(a) that the loop corresponds to a non- straight arc connecting two points of the knot. Now, in order to find a straight secant corresponding to the same arc, we have to modify the diagram as in Figure 4.4(b). If we want, we can additionally see the secant as corresponding to a crossing, just by applying a Reidemeister type II move to the diagram in Figure 4.4(b). For the general case, we can adopt the same method and modify the diagram of the trefoil to find a straight secant corresponding to the loop [x, yn]. 96 n-Essential Secants

(a) (b)

Figure 4.4: Loops corresponding to secants of the trefoil

This loop is clearly trivial in π1(Σn(K)), since by Proposition 4.16 the n-th power of each Wirtinger generator is trivial in this group. So, the associated secant is not n-essential.

We claim that these secants are strongly essential. To prove it, we just have to check that [x, yn] is non-trivial in G. In order to do so, we notice that G is the braid group on three strands B3 := hσ1, σ2|σ1σ2σ1 = σ2σ1σ2i. In order to adopt an standard notations we use σ1 and σ2 as generators of B3. For informations about braids, see [7] and [46]. For a good introduction to braids you can watch Braids. A Movie. by Ester Dalvit [16].

For braid groups, the word problem is solved. So, there exists an algorithm to check if a word is trivial or not. See [17] for a proof of the following:

Lemma 4.19. A non-trivial reduced braid, that is a braid that does not present any −1 −1 handle (i.e. a sequence of the type: σi . . . σi or of type σi . . . σi), cannot be equivalent to the trivial braid.

n −1 −n We have to check the triviality of the braids σ1σ2 σ1 σ2 in B3 for all n ∈ N. In Figure 4.5(a) is represented the one for n = 2 and in Figure 4.5(b) the one for n = 3. All the diagrams of braids are made using the Braid Applet made by Patrick Dehornoy and Jean Fromentin [18]. n-Essentiality 97

(a) (b)

2 −1 −2 3 −1 −3 Figure 4.5: The braids σ1σ2σ1 σ2 and σ1σ2σ1 σ2

There exists a well-deﬁned map f : B3 → S3, where S3 is the symmetric group in three elements. The image of a braid under f is the permutation which sends the initial three points (the ones on the left of all our ﬁgures of braids) to the end three points (in the right). If a braid is trivial, then its image under f must be the identity, i.e. the braid n −1 −n is pure. For the case where n is odd, by analyzing the diagrams, the braid σ1σ2 σ1 σ2 is not a pure braid. We conclude that for all n odd the corresponding element in G is not trivial. For the case where n is even, we apply a simple diagrammatical move in order to reduce our braid and then check its non-triviality. For the case n = 2 it is easy to check that the braid in Figure 4.5(a) is equivalent to the one in Figure 4.6.

−1 2 −1 Figure 4.6: The braid σ2 σ1σ2

Since this is a reduced braid, that is it does not present any handle, by Lemma 4.19 it cannot be trivial. n −1 −n −1 n −(n−1) More generally, the braid σ1σ2 σ1 σ2 for n even reduces always to σ2 σ1 σ2 . This can be proven diagrammatically, as in Figure 4.7, we move the violet string form the bottom to the top. −1 n −(n−1) Now, the braid σ2 σ1 σ2 is reduced and thus non-trivial again by Lemma 4.19. (Intuitively, the reduced form is such that if we take out the upper strand (the black one) we are left with a two parallel strands, that is with a trivial braid. The group formed 98 n-Essential Secants by all the braids in this form is isomorphic to the free group in two generators. The two generators can be identiﬁed with the two loop encircling the two strands. Then, any reduced word will be non-trivial.)

(a)

(b)

10 −1 −10 −1 10 −9 Figure 4.7: The braids σ1σ2 σ1 σ2 and σ2 σ1 σ2

n −1 −n n So, for all n, the braid σ1σ2 σ1 σ2 is not trivial and thus the element [x, y ] in G is not trivial and thus the corresponding secant is strongly essential.

Proposition 4.20. For any n, k ∈ N, if a secant is n-essential, then it is kn-essential.

Proof. Note that the case n = 1 or k = 1 is trivial. Let then n, k > 1. For the unbranched covers we have the following injection:

π1(X˜kn) ,→ π1(X˜n), since X˜kn → X˜n is a covering map. In the case of the branched covers we still have a map between the covers: Σkn(K) → Σn(K). However, this no longer induces an injection at the level of the fundamental groups. In fact by Proposition

4.16 we have: π1(Σn(K)) ' π1(Xñ)/Nn, where Nn is the normal group generated by the n-power of a meridian loop. All these normal subgroups of π1(Xñ) are one inside the other: Nkn < Nn. In fact we have that π1(X˜kn) < π1(Xñ) and Nkn < π1(X˜kn), so

Nkn < π1(X˜n). That is why Nkn < Nk. The only fact to notice is that, given a meridian kn loop µ, the normal subgroup generated by µ in π1(X˜kn) is contained in the normal kn subgroup generated by µ in π1(X˜n). The Double Branched Cover and its Fundamental Group 99

We prove now that kn-inessential implies n-inessential. Take the loop around the n ˜ kn ˜ secant lS , call lS its lift to Xn and lS its lift to Xkn. Then, the secant S is not kn- kn ˜ kn essential if and only if the class of lS is trivial in π1(Xkn)/Nkn, that is if [lS ] ∈ Nkn. kn n n kn ˜ ˜ But [lS ] ∈ Nkn implies [lS ] ∈ Nn, since ls is the image of ls under Xkn → Xn. Hence the secant is not n-essential. Thus, if a secant is nk-inessential it must also be n-inessential.

Take for example the case of n = 2 and n = 4. We can express the fundamental groups 4 using the Wirtinger generators. We have π1(Σ4(K)) ' π1(X˜4)/hhµ ii, an element of 4 π1(X˜4) that belongs to hhµ ii can be seen as a word corresponding to a loop with linking 2 number 0 modulo 4, is necessary trivial in π1(Σ2(K)) ' π1(X˜2)/hhµ ii.

4.3 The Double Branched Cover and its Fundamental Group

The double branched cover of knots has been extensively studied thanks to its many specific properties and its connections with different theories—like Heegaard-Floer homology, see for example [35] or [22]. This construction is also particularly relevant in our study of essential secants. There is a convenient way to form a generator set for the fundamental group of the double branched cover starting from the Wirtinger presentation of the knot group. First we define O(K), the π-orbifold group of K, yet another group that can be associated to a knot. As we shall see, this group is deeply connected with the fundamental group of the double branched cover over a knot. It has been studied extensively by Boileau and Zimmermann in [8], see also [57].

Deﬁnition 4.21. Let K be a knot. The π-orbifold group O(K) is the quotient of the knot group π1(X) by N, the normal subgroup generated by the square of a meridian loop of K:

O(K) = π1(X)/N

Remark 4.22. We have the following short exact sequence:

1 → π1(Σ2(K)) → O(K) → Z2 → 1 100 n-Essential Secants

Proof. Recall that φ2 : π1(X) → Z2 is the composition of the abelianization map and 2 the projection of Z to Z2 (Deﬁnition 4.3). Since N ' hhµ ii is a normal subgroup of 0 π1(X) that belongs to ker(φ2), we can deﬁne a map φ2 : O(K) = π1(X)/N → Z2. 0 Then, ker(φ2) = ker(φ2)/N.

By deﬁnition of covers we have: π1(X˜2) ' ker(φ2). By the previous proposition:

π1(Σ2(2)) ' π1(X˜2)/N. Thus we get: π1(Σ2(K)) ' ker(φ)/N.

0 So, we get: π1(Σ2(K)) ' ker(φ2) = ker(O(K) → Z2).

Let us consider the Wirtinger presentation of the knot group, π1(X˜2) is the subgroup of π1(X) consisting of the words of even length in the generators of this presentation.

We can form a set of generators for π1(X˜2) by selecting all the pairs on the generators of

π1(X). The relations of π1(X) identify couples of generators and thus can be interpreted as relations in this new set of generators.

Geometrically, in order to obtain the branched cover we “ﬁll” the missing solid torus.

A meridian loop in the boundary torus of X˜2 is the square of a meridian loop of X. Generally, in the n-fold branched cover, in order to obtain a closed loop we have to take n copies of a meridian loop.

3 Let K ⊂ S be a knot and D a diagram of K with n crossings. As explained in Chapter2, the Wirtinger presentation is a presentation of the knot group that can be obtained directly from the diagram D:

π1(X) = hg1, . . . , gn | r1, . . . , rni

Let I be the set of all strands of D and C the set of all crossings c = (a, b1, b2), where a, b1, b2 ∈ I. In this notation a is the over-crossing and b1 and b2 correspond to the under-crossings. We have to divide the set C of all crossings into the set C+ of positive − crossings and the set C of negative crossings. For the ﬁrst case b1 is the under-strand coming in the crossing and b2 the one going out, for negative crossings it is the opposite, see Figure 4.8. The Double Branched Cover and its Fundamental Group 101

a a b2 b2

b1 b1

Figure 4.8: A positive and a negative crossing

As usual we denote the Wirtinger generator corresponding to the strand i ∈ I by gi. Then, we can rewrite the Wirtinger presentation using this new notation:

3 π1(S \ K) = hgi, i ∈ I| gb1 ga = gagb2 ∀ (a, b1, b2) ∈ Ci

Adding to this presentation the relation which trivializes the square of one generator, or equivalently of all generators (since the generators are all conjugate to each other it makes no diﬀerence), gives us a presentation of the π-orbifold group of the knot:

2 O(K) = hg1, . . . , gn | r1, . . . , rn, g1i = 2 2 = hg1, . . . , gn | r1, . . . , rn, g1, . . . , gni = 2 = hgi, i ∈ I| gb1 ga = gagb2 for all (a, b1, b2) ∈ C, g1i

3 Now we consider the fundamental group of the double branched cover of S over K.

As before, we take the words of even length on the generators of O(K). Let xi,j be the element gigj in π1(Σ2(K)), with i, j ∈ I.

Consider the map: f : ker(φ2) → π1(Σ2(K)). This map is well deﬁned, because all the elements of ker(φ2) lift in Σ2(K) as closed loops. So, the lift in Σ2(K) will be the one going through the base-point of π1(Σ2(K)). −1 −1 −1 −1 Then note that xi,j = f(gigj) = f(gi gj) = f(gigj ) = f(gi gj ) since in π1(Σ2(K)) all Wirtinger generators are equivalent to their inverses. That is why xi,j with i, j ∈ I generates π1(Σ2(K)). 102 n-Essential Secants

π1(Σ2(K)) = hxi,j i, j ∈ I | xb1,a = xa,b2 for all (a, b1, b2) ∈ C, −1 xi,j = xj,i for all i, j ∈ I,

xi,jxj,k = xi,k for all i, j, k ∈ Ii

From this presentation we infer that xi,i = id for all i ∈ I since xi,ixi,i = xi,i. The relations of the form xa,b1 = xb2,a correspond to the Wirtinger relations. The second set of relations are the inversion rules. The last set of relations can be interpreted in π1(X) −1 in terms of couples of Wirtinger generators: gigjgj gk = gigk. This generator set can be reduced. We have the following:

Lemma 4.23. For any ﬁxed h ∈ {1, . . . , n}, the group π1(Σ2(K)) is generated by the set of n elements {xi,h}, for i = 1, . . . , n.

Proof. If we ﬁx an h ∈ {1, . . . , n}, then we can write all the generators as follows: −1 xi,j = xi,hxhj = xi,hxj,h.

Then, in order to obtain a group presentation for π1(Σ2(K)) we add the relations which correspond to the Wirtinger relations.

Given a knot diagram, at each crossing three arcs converge. Recall Deﬁnition 2.28: The loop around the crossing c = (a, b , b ) is l = g g−1, independent of the sign of the 1 2 c a b2 crossing, see Figure 4.9. We call the element xc ∈ π1(Σ2(K)) the image of lc under the map f deﬁned above.

a g g -1 b a b2 a 2 b2

g g -1 a b2 b1 b1

(a) (b)

Figure 4.9: The loops around a positive and a negative crossing The Double Branched Cover and its Fundamental Group 103

Lemma 4.24. Let c = (a, b1, b2) be a crossing in a diagram D of a knot K. Every loop xi,j ∈ π1(Σ2(K)) formed by the product of two Wirtinger generators or their inverses among ga, gb1 and gb2 can be expressed in terms of the loop xc around the crossing c.

Proof. In the previous notation, consider a crossing c = (a, b1, b2) ∈ C. Let us analyze what kind of loops there are around a crossing. For every c =

(a, b1, b2) ∈ C three arcs meet, so there are nine diﬀerent loops around two arcs because the orientation does not matter: xa,a, xa,b1 , xa,b2 , xb1,a, xb1,b1 , xb1,b2 , xb2,a, xb2,b1 , xb2,b2 . Three of these are trivial in the fundamental group of the double branched cover: xa,a, xb1,b1 and xb2,b2 . We have deﬁned l = g g−1 in π (X). Now, we consider the map f as before f : c a b2 1 ker(φ2) → π1(Σ2(K)). As we have seen, the kernel of f is N, the normal subgroup of ker(π1(X) → Z2) = ker(φ2) generated by the square of a meridian. Consider then the image of x under f. We have f(l ) = f(g g−1) = f(g g ) = x . c c a b2 a b2 a,b2 The following equalities hold for all crossings:

 xa,a = xb1,b1 = xb2,b2 = id   x = x = x  a,b2 b1,a c  −1 xa,b1 = xb2,a = xc   x = x2  b1,b2 c   −2 xb2,b1 = xc

The ﬁrst line expresses the fact that in π1(Σ2(K)) all squares of Wirtinger generator are trivial. The second line is a consequence of the relation in π1(Σ2(K)) corresponding to the Wirtinger relation. The ﬁrst equality of the third line is also a consequence of the

Wirtinger relation and the second equality derives from the fact that in π1(Σ2(K)) we −1 always have xi,j = xj,i. 2 Moreover we have: xb1,b2 = xc since xc = xa,b2 = xb1,a and xb1,b2 = xb1,axa,b2 . For similar reasons the last line also holds.

Therefore we can express any of these nine loops as powers of xc = xa,b2 .

Lemma 4.25. The two loops the Figure 4.10 lift to homotopic loops in the double branched cover. 104 n-Essential Secants

a a b2 b2 x1 x'1

b1 b1

x2 x'2

Figure 4.10: Equivalent loops in Σ2(K) for a positive (left) and a negative crossing (right)

Proof. This is a consequence of the last proposition. For a positive crossing, we have that in π (X), x = g−1g lifts to a loop homotopic to 1 1 b1 a x2 = gb1 ga in Σ2(K). Recall in fact that in Σ2(K) the lift of the square of any Wirtinger generator is trivial, we have: f(x1) = f(x2). 0 For negative crossings we have a symmetric situation: x1 = gb1 ga which lifts to a loop homotopic to x0 = g−1g in Σ (K). 2 b1 a 2

Proposition 4.26. Every loop xi,j ∈ π1(Σ2(K)) can be expressed as a product of loops around crossings of D.

Proof. Let us consider the graph G associated to the knot diagram D in the following way: To the crossings we associate vertices disregarding the relative height information. In this way we get a 4-valent graph from D. Every edge—going from one vertex to another—will correspond to a part of an arc in D (note that an arc in a knot diagram is a strand going from an under-crossing to another under-crossing). An arc will correspond to as many edges as the crossings it goes through, plus one, see Figure 4.11.

(a) (b)

Figure 4.11: A knot and its associated graph The Double Branched Cover and its Fundamental Group 105

We assign to the edges of G the same name as the loops in the Wirtinger presentation associated to the corresponding arcs. Therefore there will be diﬀerent edges with the same label.

Now, consider two diﬀerent edges, say gi and gj. There is always a ﬁnite path connecting them, gi = g0, g1, . . . , gd = gj, where gk+1 is incident to gk. In fact we are dealing with a knot and therefore our graph is connected.

Consider now the loop xi,j = f(gigj), which corresponds to a product of two Wirtinger generators. Remember that f is the map deﬁned above from ker(φ2) to π1(Σ2(K)). We ± claim that this loop can be decomposed as a product of d + 1 loops of the form xci with = 0, 1, 2. Let us now see how.

We can follow the path g0 = gi, g1, . . . , gd = gj. This path consists of edges that meet in a sequence of d crossings. Call them c1 . . . cd.

By Lemma 4.24 we have that gkgk+1 for k ∈ {0, . . . , d − 1} are equivalent to loops around crossings because they are products of loops around adjacent edges. We get:

−1 −1 −1 gigj = (gig1)(g1 g2)(g2 g3) ··· (gd−1gj)

Each of the couples appearing is a power of a loop around a crossing: x± with ck = 0, 1, 2, k ∈ {1, . . . d + 1}.

Thus, we have managed to write an arbitrary element of π1(Σ2(K)) as a product of loops around crossings.

Corollary 4.27. Let D be a diagram for the knot K. Then the group π1(Σ2(K)) is generated by all the loops around the crossings of D.

We can reduce the set of generators keeping just one for every band, which is a sequence of 2-sided regions in a knot diagram, see Figure 4.12.

Figure 4.12: A band in a knot diagram

That is, using the following lemma, we decrease the number of generators by one for each 2-sided region. 106 n-Essential Secants

Lemma 4.28. The loops in Σ2(K) corresponding to the crossings of the same band are equivalent.

Proof. There are two different possibilities for the loops around the crossing forming a band. They can be vertical or horizontal, as in Figure 4.10. In the first case they are equivalent since we can slide the loop around one crossing along the band and make it become a loop around any other crossing of the same band. The second case, where the loops are horizontal, can be reduced to the first one thanks to Lemma 4.25.

Figure 4.13: Loops in a band

Proposition 4.29. The 2-essentiality of crossings is preserved by ﬂypes.

Proof. As we know, performing a flype to a knot diagram does not change the corresponding knot type. All the crossings except one remain the same and their essentiality remains unchanged. In fact, the essentiality of the crossings inside the “box” which is flyped, the one containing the F in Figure 4.14, is not changed. Neither is the essentiality of the crossing outside the local part of the diagram shown in Figure 4.14, where the modification occurs. In fact, there is a one to one correspondence between the crossings of the diagram before and after the flype: The crossings inside the box correspond to crossings inside the flyped box, the crossing outside on the right of the box will correspond to the crossing on the left and the crossings outside correspond to themselves. To prove that the 2-essentiality of the crossing outside the box that is switched is also preserved, we use once again Lemma 4.25. In fact, if we consider a vertical loop around the crossing outside the box that is flyped, then it is easy to see that this is equivalent to the loop around that same crossing after having performed the flype. We just have to to slide it around the box. The case of horizontal loops can be reduced to this one. 2-Essentiality, a Combinatorical Approach 107

F F

Figure 4.14: A ﬂype

Sometimes local modiﬁcations in a knot diagram can change the essentiality of crossings “far away”. For example if we change a crossing in an unknotting number one knot diagram, and we transform the diagram to a diagram of the unknot, all the crossings that previously where essential become inessential.

Corollary 4.30. Let K be a knot and D a minimal diagram of it. If all the crossings of D are 2-essential, then all the crossing of any minimal diagram of K are 2-essential.

Proof. The claim follows from the previous lemma and the fact that any two minimal diagrams of the same knot can be related by ﬂypes (Flype Conjecture, Theorem 3.1).

4.4 2-Essentiality, a Combinatorical Approach

In this section we prove that any diagram of a non-trivial knot must have at least three 2-essential crossings (Theorem 4.42). In order to do so we ﬁrst need the following preliminary facts. From here on we will consider the combinatorial structure associated to knot diagrams: We will analyze the 4-valent graphs obtained from a knot diagram forgetting the relative height information at the crossings, as we did in the proof of Proposition 4.26. In doing so we consider the regions in which the plane is divided, that is, the faces of the associated graph. If the vertex A belongs to the boundary of the region R, we will write the short notation: A ∈ R.

Remark 4.31. All crossings in a knot diagram belonging to 1-sided regions are inessential. In fact, there exists an embedded disk bounded by the secant and the part of the knot corresponding to the 1-sided region. 108 n-Essential Secants

Remark 4.32. The two crossings belonging to the same 2-sided regions are either both 2-essential, or both 2-inessential. A 2-sided region is in fact a band. Thus the statement follows from Lemma 4.28.

Remark 4.33. All crossings in a knot diagram belonging to a tree ending in nugutory crossings, like the one in Figure 4.15, are inessential.

Proof. We can now start noticing that the most external crossings are inessential, but then we can “undo” them and obtain another graph corresponding to a diagram that represents the same knot type. Then we apply the same process to all crossings.

Figure 4.15: A tree of regions ending in nugutory crossings

Lemma 4.34. If a crossing in a knot diagram is strongly essential, then it must belong to four distinct regions.

Proof. A crossing in a knot diagram meets four regions. As we know, every knot diagram admits a checkerboard coloring (see Remark 3.34), and thus it will be impossible for adjacent regions to be equal—one will be black and the other white. Thus the four regions around a crossing are at least divided in two groups, the white and the black ones. We claim that if the two opposite regions around a crossing are equal, then the diagram is not reduced and the crossing is not strongly essential. Let A1 and A3 be two points close to the crossing and belonging to opposite regions, like in Figure 4.16. If these opposite regions were equal, there would exist a path that does not intersect the knot and connects A1 and A3. Suppose now to connect A1 to A3 by another path going through the crossing. Then, if we join these two points with the two paths (the one through the crossing and the one that does not intersect the knot), we obtain an 2-Essentiality, a Combinatorical Approach 109 embedded circle intersecting the knot in just two points at the crossing. This implies that the crossing is nugutory and the diagram is not reduced. By Corollary 2.41 the crossing is thus not strongly essential.

Figure 4.16: Two opposite regions are equal

Let A and B be two crossings in a diagram D for some non-trivial knot K. Let R1,

R2, R3 and R4 be the regions around A and x1, x2, x3 and x4 be the edges that go out of A, as in Figure 4.17.

x2 x4 R3 B R4 R2 R1 x1 x3

Figure 4.17: The regions around a crossing

Lemma 4.35. A knot diagram of a non-trivial knot contains a 2-essential crossing.

3 Proof. By Theorem 4.10 Σ2(K) is not homeomorphic to S and thus π1(Σ2(K)) is not trivial. Therefore in any presentation of this group there must be non-trivial generators. Since by Corollary 4.27 we can compute a presentation whose generators correspond to the crossings of the diagram D, there must be at least one 2-essential crossing c.

Lemma 4.36. A region of a graph corresponding to a knot diagram cannot contain a single 2-essential crossing. 110 n-Essential Secants

Proof. Let c be a 2-essential crossing in a knot diagram D and let R be a region containing c. The crossing c cannot be the only 2-essential crossing of R. In fact by Lemma 2.25 we have a relation for each region of D; if c were the only non-trivial loop, then the relation for R would look like: c id ··· id = id, the non-triviality of c would be contradicted. So, if a region of D contains a 2-essential crossing, then it must contain at least another one.

Lemma 4.37. Two distinct crossings in a prime knot diagram that share two distinct adjacent regions have an edge connecting them.

Proof. Suppose A and B are two crossings sharing two adjacent regions (note that since they are adjacent they must be distinct as well). Since we are working with diagrams on 2 S , all regions are bounded. We can therefore assume that the two regions shared by A and B are adjacent and bounded.

Let B belong to the boundary of both R2 and R1, see Figure 4.19. Then, we claim that the edge x1 must connect A to B. If this were not the case, following the boundary of the region R1 from x1 to an edge connected to B and again from x1 along the boundary of R2, we would obtain a part of the diagram connected with the rest by just x1 and the edge going out from B between R1 and R2. Then the graph would correspond to a composite knot diagram. In fact, in this case we could identify a circle intersecting the knot diagram in just two points and such that the parts in which the knot is divided are both non-trivial arcs, see Figure 4.18.

x2 x4 R3 B R4 R2 R1 x1 x3

Figure 4.18: A composite diagram

That is why, if we are dealing with a prime diagram, the edge x1 is connecting A to B, Figure 4.19. 2-Essentiality, a Combinatorical Approach 111

x2 x4 R3 B R4 R2 R1 x1 x3

Figure 4.19: The edge AB

Lemma 4.38. Two distinct crossings in a prime knot diagram that share three distinct regions have two edges connecting them, and thus between these edges there is a 2-sided region.

Proof. We apply the previous lemma twice. Let A, B ∈ R1,R2,R3. Since A, B ∈ R1,R2, x1 is an edge connecting A to B. Since A, B ∈ R2,R3, also x2 is an edge connecting A to B. This conﬁguration forms a 2-sided region, as in Figure 4.20.

x2 x4 R3 B R4 R2 R1 x1 x3

Figure 4.20: Two vertices share three regions

Lemma 4.39. Two distinct 2-essential crossings in a prime knot diagram cannot share all the four distinct regions to which they belong.

Proof. Call the two crossings A and B and suppose by contradiction they were sharing all the four regions. 112 n-Essential Secants

As before, call the regions to which A belongs, R1, R2, R3 and R4, as in Figure 4.17. Then B shares all these regions as well. By the previous lemmas, if we consider that

A and B share R1,R2 and R3, we have that x1 and x2 must be edges connecting A to

B. But A and B also share R3,R4 and R1, and thus x3 and x4 are also edges connecting A to B. This implies that our graph, corresponding to the knot diagram is the following:

Figure 4.21: Two crossings sharing all the 4 regions

But this is impossible. In fact in this case we would have just two crossings and no graph with only two vertices can correspond to a diagram of a non-trivial knot. Moreover, the graph in Figure 4.21 can only correspond to a link diagram, not to a knot diagram.

Now we present another well-known application of the theorem of Van Kampen:

Lemma 4.40. Let K1 and K2 be two knots. Then:

π1(Σ2(K1#K2)) = π1(Σ2(K1)) ∗ π1(Σ2(K2)).

Proof. By Proposition 4.12 we know that the double branched cover of a composite knot is the connected sum of the double branched covers of the two components: If

K = K1#K2, then Σ2(K) ' Σ2(K1)#Σ2(K2). Thus, we can apply once again the theorem of Van Kampen (Theorem 5.18) to calculate the fundamental group of Σ2(K). ◦ We have to chose appropriate open sets whose union is Σ2(K). Let U :=Σ2(K1) and ◦ ◦ 2 V :=Σ2(K2). Then, we glue U and V along S × I, so that they overlap and we can apply the theorem. We have the following commutative diagram:

i j 11 U PPq1 2 × I Σ2(K) S PPq 1 i2 V j2 2-Essentiality, a Combinatorical Approach 113

◦ 2 2 Since π1(U) = π1(Σ2(K1)), π1(V ) = π1(Σ2(K2)) and π1(S × I) = π1(S ), the induced maps at the level of the fundamental groups are the following:

π1(Σ2(K1)) j i∗1 : X ∗1 Xz {id} X π1(Σ2(K)) XXXz : i∗2 j∗2 π1(Σ2(K2))

We have π1(Σ2(K)) = π1(Σ2(K1))∗π1(Σ2(K2)), thus both π1(Σ2(K1)) and π1(Σ2(K2)) inject into π1(Σ2(K)).

Corollary 4.41. Let D = D1#D2 be a composite diagram of K = K1#K2, where D1 is a diagram of K1 and D2 a diagram of K2. Then a crossing of D is 2-essential if and only if it is 2-essential as a crossing of Di, for i = 1 or 2.

Theorem 4.42. Any diagram of a non-trivial knot must have at least three 2-essential crossings. This limit is sharp: There exists an inﬁnite family of diagrams with only three 2-essential crossings.

Proof. By Lemma 4.35, there must be at least one 2-essential crossing, call it A. As we have seen, each crossing in a knot diagram can be seen as belonging to four regions. For a 2-essential crossing these must be all distinct. So, for any of the four distinct regions around A at least another crossing must be 2-essential (Lemma 4.36). Therefore there must exist B, another 2-essential crossing. Let us ﬁrst consider prime diagrams, and extend later the result to any kind of diagrams. By Lemma 4.39, we know that A and B cannot share all of them. Thus there must be a region to which just one of these two crossing belongs. Call it R. As before, a region cannot contain just one 2-essential crossing, and thus there must exist in R another 2-essential crossing that is distinct from A or B. So, the claim holds for prime diagrams. But if a diagram is not prime, it is formed by prime-components (Proposition 1.45). For each of the components we can apply the result concerning prime diagrams. In fact, by Lemma 4.40 we know that if a crossing of one component is 2-essential, it remains so as crossing of the composite diagram. Thus, for each non-trivial component we have at least three 2-essential crossings. 114 n-Essential Secants

This limit of three crossings is sharp for non-prime diagrams since it is realized by diagrams of the trefoil knot with extra nugutory crossings. More in general it is also realized by composition of non-trivial diagrams of the unknot with the trefoil knot. These are diagrams with an arbitrary high number of crossings but with just three 2- essential crossings, see Figure 4.22. Note however that in the case of prime diagrams this phenomenon can no longer appear, as we shall later see.

Figure 4.22: Diagram of the trefoil knot with three 2-essential crossings and one inessential crossing

For prime diagrams with small crossing number we have the following:

Proposition 4.43. In a prime diagram of a non-trivial knot with four or ﬁve crossings, all the crossings are 2-essential.

Proof. By the classiﬁcation of 4-valent graphs in the sphere corresponding to knots, any such diagram is the same graph as one corresponding to a minimal diagram of the trefoil, to a minimal diagram of ﬁgure-eight knot, or to one of the two 5-crossings knots, see Figure 4.23.

(a) (b) (c)

Figure 4.23: Graphs corresponding to prime knot diagrams 2-Essentiality, a Combinatorical Approach 115

From the proposition above, we have at least three 2-essential crossings. As before, we consider the generators of π1(Σ2(K)) as corresponding to crossings of the diagrams.

For the case of n = 4 the crossings are pairwise belonging to the same 2-sided region, and thus if 3 are 2-essential, the fourth must be also.

For the case of n = 5, in the ﬁrst graph (Figure 4.23(b)) all crossings belong to the same band and thus are all equivalent.

In the second graph (Figure 4.23(c)) there are two bands. At least one crossing must be 2-essential, so at least one band is formed by 2-essential crossings. Suppose the “upper” band in Figure 4.24 is formed by 2-essential crossings. Then, by applying Lemma 4.36 to

Region R1, also the crossings of the other band must be 2-essential. If we suppose that the crossings of the lower band are 2-essential, we apply again 4.36, this time to Region

R2, and we ﬁnd that the other ones must also be 2-essential. In fact, this can be seen easily from Figure 4.24 using Lemma 4.36.

Figure 4.24: Regions in a graph corresponding to a ﬁve crossing prime knot diagram

See for example in Figure 4.25 diagrams of the trefoil with three, four or ﬁve 2-essential crossings. Here, by the previous proof, all the crossings are 2-essential. 116 n-Essential Secants

(a) (b) (c)

Figure 4.25: Diagrams of the trefoil with three, four and ﬁve 2-essential crossings

We conjecture that in any prime diagram of any non-trivial knot there must be at least five 2-essential crossings. However, for more than five crossings, we cannot use the same approach. For example, the following figure depicts an infinite family of 4-valent graphs corresponding to knot diagrams that might only have five 2-essential crossings. As we did before, applying Lemma 4.36 to different regions, we obtain that the marked crossings in Figure 4.26 must be 2-essential, but that the others may also not be 2-essential.

n-5

Figure 4.26: Graph corresponding to a non-trivial knot that might only have ﬁve 2- essential crossings Chapter 5

2-essentiality and Families of Knots

In this chapter we focus on 2-essentiality analyzing the crossings of particular diagrams of different classes of knots. More specifically, we first consider rational knots and then torus knots of types T (2, q), T (3, q) and T (5, q). We will show that in a minimal diagram of a rational knot, all crossings are 2-essential. Moreover, we show examples of crossings in non-minimal diagrams of rational knots that are strongly essential but not 2-essential. Then, we will show that also in the standard braid diagrams of the torus knots of the mentioned types all crossings are 2-essential. Note that this last result concerns also non-alternating knots.

5.1 Rational Knots

In this section we prove that all crossing in a minimal diagram of a rational knot are 2-essential. First we consider the standard 4-plat diagram that can be associated to a rational knot, and then we use the fact that any two minimal diagrams of an alternating knot can be related by a sequence of flypes (Theorem 3.1). Rational knots are also called 2-bridge knots because they are the only knots whose bridge number is two, i.e. seen as curves in space, they can be isotoped to curves whose number of maxima (or minima) for the standard height function is two. The name Rational Knot was first used by John Conway who identified this special class of knots

117 118 2-essentiality and Families of Knots with the closure of rational tangles and identiﬁed a correspondence between these and rational numbers, via continued fractions. See [14, Ch. 8] for a good explanation of this correspondence.

5.1.1 Conway Normal Form

In this section we state some well-estabilisched facts about rational knots. We will deﬁne them through the Conway normal form, adopting the notation of [30, Ch. 2].

Deﬁnition 5.1. A rational link is a link that admits a diagram equivalent to one of the diagrams depicted in Figure 5.1, where ai denotes the number of crossings of every

“band”. The ai are integers, the corresponding crossings will be positive or negative crossings (in the ﬁgure we depicted the diagram corresponding to positive ai). This is called a 4-plat diagram and it is denoted by C(a1, . . . an).

a1 n odd an

a1 n even

a2 an

Figure 5.1: The diagram C(a1, . . . , an)

A rational knot is a rational link of one component.

To each 4-plat of a rational link, we can naturally associate a continued fraction. This is a well-known construction, see [14, Ch.8].

Deﬁnition 5.2. Let (a1, . . . an) be a n-uple of rational numbers with an 6= 0. Then, the associated continued fraction is:

1 [a1, . . . an] := a1 + a + ... 1 2 an Rational Knots 119

Continued fractions can be evaluated and therefore correspond to rational numbers. On the contrary, each rational number admits, by the Euclidean algorithm for division, a continued fraction. For α and β relatively prime integers we have: r−1 := α, r0 := β.

We start with i = 1 and for each step i = i + 1, we ﬁnd a quotient qi and a remainder ri < ri−1 such that: ri−2 = qiri−1 + ri. The remainders decrease at each step but can never be negative. For some n a remainder rn will be equal zero, at which point the algorithm stops. We have α/β = [q1, . . . qn].

For a continued fraction [a1, . . . , an] we have the following partial relations, call Nk respectively Dk the numerator, respectively denominator, of the the evaluation of the continued fraction [a1, . . . ak], for k < n: Nk/Dk = [a1, . . . ak]. Then, the following holds:

Nk = akNk−1 + Nk−2, with N−1 = 1 and N−2 = 0

Dk = akDk−1 + Dk−2, with D−1 = 0 and D−2 = 1 However, the same rational number correspond to more than one continued fraction.

For example, if an is strictly positive, then [a1, . . . an] = [a1, . . . an − 1, 1].

The continued fraction associated to the diagram C(a1, . . . , an) is [a1, . . . , an]. Since the evaluation of a ﬁnite continued fraction is a rational number, we can always ﬁnd two co-prime integers α, β such that α > 0 and such that α/β = [a1, . . . , an].

This gives us a natural way to associate to the diagram of type C(a1, . . . , an) the couple (α, β). This couple is named after Murasugi [45] the type of the corresponding rational link, this deﬁnition is useful since we have the following well known correspondence, see [45, Ch. 9] :

Theorem 5.3. Let C(a1, . . . , an) and C(b1, . . . , bn) be two 4-plat diagrams. If their type is the same, then they are diagrams of equivalent rational links.

Deﬁnition 5.4. Let α and β be two coprime integers such that α > 0. Call R(α, β) the link of type α/β.

Two links R(α, β) and R(α0, β0) are equivalent if and only if α = α0 and β = β0, (mod α) or β−1 = β0, (mod α). This is an important result [45, p. 189] whose proof relies on the correspondence between knots and lens spaces estabilisched by Schubert. The double 3 branched cover of S over the rational link of type R(α, β) is L(α, β), see [55]. 120 2-essentiality and Families of Knots

In the following we state more facts about rational links that we will use in the later discussion.

Proposition 5.5. The link R(α, β) is a knot if and only if α is odd.

Proof. The result follows from the properties of the determinant and the fact that the determinant of R(α, β) is α, see [29].

The following is another well-known fact about rational knots:

Proposition 5.6. Rational knots are alternating.

Proof. Let R(α, β) be a rational knot. A diagram of R(α, β) is C(a1, . . . , an), where

[a1, . . . , an] = α/β. If β is positive, then we can choose a continued fraction for that evaluate α/β with all the ai positive (by the Euclidean algorithm of division). Analogously, if β is negative, then we can choose a continued fraction that evaluate α/β with all the ai negative.

By construction, a diagram of type C(a1, . . . , an) is alternating if and only if all the ai have the same sign. So, there is always an alternating diagram representing the rational knot R(α, β).

Yet another important property of rational knots, which also derives from the correspondence between them and lens spaces (see [54]), is the following:

Proposition 5.7. Rational knots are prime.

Deﬁnition 5.8. We denote with D(α, β) a representative diagram of a rational knot

R(α, β), if the following holds D(α, β) = C(b1, . . . , bn) such that:

• all bi have the same sign as β,

• b1 and bn are diﬀerent from ±1.

Kawauchi [30, p. 25] proves that a representative diagram always exists for any rational knot and it is unique up to the following relation: Two representative diagrams 0 0 0 0 D(α, β) = C(b1, . . . , bn) and D(α , β ) = C(b1, . . . , bn0 ) represent the same rarional link 0 0 n−1 if and only if n = n and bi = bi for all i or bi = (−1) bn−i for all i. Rational Knots 121

By construction, the diagram D(α, β) is an alternating reduced diagram of the knot R(α, β), thus it is minimal.

We assume in the following that β is a positive integer. Therefore all the ai will be positive; the case where is negative β is analogous.

a1 n odd an

a1 n even

a2 an

Figure 5.2: The diagram C(a1, . . . , an)

As we have already mentioned, Schubert [55] proved that the two-fold branched cover of the three-sphere along a rational knot is a lens space. This achievement lead to a classiﬁcation of rational knots and it is a key-result for our study of the 2-essentiality of crossings in a diagram of a rational knot. We have the following one to one correspondence:

{Rational links} / knot type ↔ {Lens spaces} / homeomorphism

The fundamental group of the lens space (p, q) is the cyclic group of order p:

π1(L(p, q)) = Zp

The correspondence above is relevant in this context because in order to prove the 2-essentiality of secants and crossings, we need to prove the non-triviality of certain elements in the fundamental group of a cyclic branched cover over the knot. In what follows, we will prove that in a minimal diagram of a rational knot all the crossings are 2-essential. Note that since rational knots are prime and alternating, by Lemma 3.44, we already know that all the crossings in a minimal diagram of a rational knot are strongly essential. Thus this new result about 2-essentiality strengthens the previous one. 122 2-essentiality and Families of Knots

5.1.2 2-essential Crossings

Let us now consider the diagram C(a1, . . . , an) in Figure 5.2 corresponding to the knot α R(α, β), where β = [a1, . . . , an]. As we have seen through the construction, to each ai corresponds a “band”: A sequence of ai crossings forming 2-sided regions.

Consider now a loop xi around each of these bands, with the following convention: First we go around the strand below (which can be a under- or over-strand) and then around the one above, like in Figure 5.3. These loops are in our notation powers of lc, the standard loop we consider around a crossing c.

a1 n odd an

a1 n even

a2 an

Figure 5.3: Loops around crossings in the diagram C(a1, . . . , an)

For our purposes, the loops corresponding to diﬀerent crossings belonging to the same band are always equivalent and therefore we need to analyze just one of them. This is because we consider them in π1(Σ2(K)), see Lemma 4.28.

Lemma 5.9. Let xi be the class of a loop corresponding to the i−th band of the diagram

C(a1, . . . , an) in π1(Σ2(K)). Then, every xi is a power of x1. More precisely, xi =

N[a1,...,ai−1] x1 , where N[a1, . . . , ai] is the numerator of the continued fraction [a1, . . . , ai].

Proof. These loops can be interpreted as elements of the fundamental group of the cyclic branched cover over the rational knot, in fact they are a product of two Wirtinger generators or their inverses. To prove this result we will use Lemma 2.25 to every region of D. The diagram in Figure 5.2 presents n + 2 k-sided regions, with k > 2. Of these, n are subdivided in the two types represented in Figure 5.4. Rational Knots 123

xi+1

xi xi+2

(a) i even

xi xi+2

xi+1

(b) i odd

Figure 5.4: Region types

To have a uniform notation, we introduce two trivial loops x0 and xn+1 at the beginning and at the end of the diagram.

As we have seen, considering all the xi as elements of π1(Σ2(K)), each “vertical” loop will be equivalent to a “horizontal” one, by Lemma 4.28. Using the previous observation, we obtain the following relations from Figure 5.4:   ai+1 −1 xixi+1 xi+2 = id for i even  ai+1 −1 xi+1 xixi+2 = id for i odd

That is:

  ai+1 xi+2 = xixi+1 for i even  ai+1 xi+2 = xi+1 xi for i odd

By induction on i, they are all powers of the same element.

ai+1 Therefore xi+1 and xi+2 commute. So, we will have the same expression for i odd ai+1 or even: xi+2 = xi+1 xi, that is:

ai−1 xi = xi−1 xi−2 124 2-essentiality and Families of Knots

Let us consider which power of x1 we need in order to obtain xi, we call ki the exponent such that:

ki xi = x1

We have the initial conditions, k0 = 0 and k1 = 1 and the following relations:

ki = ki−1ai−1 + ki−2

This is exactly the sequence that determines the numerator of the continued fraction

[a1, . . . , ai−1]. In fact, as mentioned earlier, given a continued fraction [a1, . . . an], the numerators of the continued fraction [a1, . . . , ak] are determined by the following formula:

Nk = akNk−1 + Nk−2, with N−1 = 1 and N−2 = 0.

N[a1,...,ai−1] N[a1,...,an−1] Then we have xi = x1 , and at last xn = xn .

Theorem 5.10. All crossings in the diagram D(α, β) are 2-essential.

Proof. We saw in Lemma 5.9 how to express every loop corresponding to a crossing as a power of the loop x1 corresponding to the ﬁrst band of the diagram C(a1, . . . , an). Now, to check the 2-essentiality of a crossing means to check the triviality of the corresponding loop in π1(Σ2(K)). The two-fold branched cover over R(α, β) is the lens space L(α, β). The fundamental group of this space is the cyclic group Zα.

We claim that the loop x1 cannot be trivial. By the previous lemma and by Corollary p 4.27, it is in fact a generator of π1(Σ2(K)). Then, also x1 with p relatively prime with α will be a generator of π1(Σ2(K)). q All elements of the form x1 will be non-trivial if q is not a multiple of α, so in particular for all q < α. Since we are considering a presentation with all the ai strictly positive, we have that N[a1, . . . , ak] < N[a1, . . . , an] = α for all k < n.

Therefore using the expression for xn:

N[a1,...,ai−1] xi = x1 we get the non-triviality of each xi.

Corollary 5.11. All crossings of a minimal diagram of a rational knot are 2-essential. Rational Knots 125

Proof. Rational knots are alternating (Proposition 5.6), therefore any two minimal diagrams of the same rational knot can be related by a sequence of ﬂypes (Theorem 3.1). Since by Proposition 4.29 the 2-essentiality of crossings is preserved by ﬂypes, the result follows from the previous theorem.

5.1.3 Non-minimal Diagrams

Let us now investigate the nature of the crossings of some non-alternating (and thus non-minimal) diagrams of rational knots. We show here examples of families of rational knot diagrams containing crossings that are not 2-essential but are strongly essential.

Proposition 5.12. The following families of diagrams of rational knots have crossings that are not 2-essential:

• C(y, −1, 1, 2(y − 1)), y ∈ N ∪ {0}

• C(z, 1, −1, 2(z + 1)), z ∈ N ∪ {0}

Figure 5.5: The diagram C(4, −1, 1, 6)

Proof. In these cases we have that N2 = −N4 (remember that Nk is the numerator of the continued fraction [a1, . . . , ak] ). In general, for a continued fraction [a1, . . . , an] we have N2 = a1a2 + 1 and N4 = a1a2a3a4 + a1a2 + a1a4 + a3a4 + 1.

For the ﬁrst family we have: N2 = −y + 1 and N4 = −2y(y − 1) − y + 2y(y − 1) + 2(y − 1) + 1 = y − 1.

For the second family we have: N2 = z + 1 and N4 = −2z(z + 1) + +z + 2z(z + 1) − 2(z + 1) + 1 = −z − 1.

So in both cases we have that N2 = −N4. 126 2-essentiality and Families of Knots

From Lemma 5.9 we have:

Ni−1 xi = x1 where xi is the loop corresponding to the i-th band in the diagram C(a1, a2, . . . , an), which is associated to the continued fraction [a1, . . . , an].

The fundamental group of Σ2(K) is Zα, where α = N4. This is why, if N2 = −N4, N2 the elements x3 = x1 in the previous three cases are trivial.

5.2 The Torus Knots T (2, q), T (3, q) and T (5, q)

As we have seen, torus knots can be identiﬁed with a pair of coprime integers. The knot T (p, q) can be isotoped as to lie in a standard torus and encompass it p times in one direction and q times in the other. Torus knots are often represented in the form of braids, as in Figure 5.6, the corresponding braid word being: q (σ1, σ2, . . . , σp)

In the diagram we only see a part of the knot. We have to imagine that each strand on the right at a certain height is connected to the one on the left at the same height by unknotted an unlinked arcs, this is the standard circular closure of a braid.

Figure 5.6: Braid diagram of the torus knot T (3, 7)

We have already analyzed (2, q) torus knots and proved that in a minimal diagram of any such knot all the crossings are strongly essential. Now we will prove the stronger result that all the crossings in such diagrams are 2-essential (by Lemma 3.42, we can analyze only the standard braid diagram of such knots without loss of generality). Moreover, we will treat the case (3, p) and (5, p). If we limit our domain to these families of torus knots, it is possible to use combinatorial arguments to prove that in a standard braid diagram of them, all crossings have to be 2-essential. For these families of knots we have the following special property: The Torus Knots T (2, q), T (3, q) and T (5, q) 127

Theorem 5.13. Any crossing of a braid diagram of the torus knots T (2, q), T (3, q) and T (5, q) for all q, is 2-essential.

Proof. Let us ﬁrst look at the braid diagrams of the torus knots of type T (2, q). Since these diagrams present a single “band”, either all the crossings are 2-essential or all the crossings are not 2-essential (Lemma 4.28). By Lemma 4.35, there is at least a 2-essential crossing. Therefore all crossings are 2-essential.

Let us now consider the knots T (3, q), see Figure 5.6.

The fact that all the crossing have to be 2-essential can be inferred by reasoning on the braid diagrams of such knots: Exploiting the symmetries of these diagrams, we can “slide” a loop around a crossing to any other loop around a crossing. That is, they are in the same free homotopy class of X, being conjugate to one another. The loops in Figure 5.7 are all free homotopic to each other. The ﬁrst two are actually homotopic to each other, the third is instead a conjugate and so on. Actually all the loops around a crossing free homotopic to each other, it is enough to use the fact that the strands on the right are actually connected to the ones on the left. So we carry our loop “around” the braid diagram.

Figure 5.7: Equivalent loops around crossings in the diagram of T (3, 7)

This means that these loops are equivalent: If one is 2-essential, then they all have to be so. (Note that the same holds for strong essentiality).

We know by Lemma 4.35 that there is at least a 2-essential crossing, therefore they all must be so.

For the torus knots of type T (5, q) we have two diﬀerent free homotopy classes of loops around crossings: 128 2-essentiality and Families of Knots

Figure 5.8: Braid diagram of the torus knot T (5, 6)

The loops l1 and l2 in Figure 5.9 are equivalent since we can slide one into the other.

The loop l3 is still conjugate to the previous ones: when we slide it we have to go “under” other strands of the diagram a certain number of times. Thus these three loops are in the same free homotopy class of X.

Figure 5.9: Equivalent loops around crossings in the braid diagram of T (5, 6)

The same reasoning can be applied for all the loops corresponding to all upper and lower crossings of this diagram, they are all conjugate to each other, see Figure 5.10.

Figure 5.10: Equivalent crossings in the braid diagram of T (5, 6)

If we look at any other crossing, we ﬁnd that they also form a class of free homotopy loops, see Figure 5.11. The Torus Knots T (2, q), T (3, q) and T (5, q) 129

Figure 5.11: Equivalent crossings in the braid diagram of T (5, 6)

Now we can apply Lemma 4.36 to the regions of the diagram. Call x1 the free homotopy class of the loops around the upper and lower crossings, and x2 the other class (corresponding to the crossings in “the middle” of the diagram). Now, as we have seen, if one of the crossings of one class is trivial, then they all must be so. So, either they are all 2-essential, or all not 2-essential. We know that there is at least one 2-essential crossing (Lemma 4.35). Suppose ﬁrst that the crossings corresponding to the class x1 are all 2-essential. Then, applying Lemma

4.36 to region R2 in Figure 5.12, we have that also the crossings corresponding to the class x2 must all be 2-essential. On the other hand, if we suppose that the crossings corresponding to the class x2 are all 2-essential, we can apply the same reasoning to region R1, and infer that the one corresponding to x1 must also be 2-essential.

Figure 5.12: Regions in the diagram of T (5, 6)

So, we have shown that all crossings have to be 2-essential. We reasoned using a particular knot: T (5, 6). Nevertheless, the same holds for any other torus knot T (5, q), with q > 5 relatively prime with 5. In fact, none of the steps in our reasoning relied on the particular “length” of the braid diagram. The only important thing to notice is that when we analyze a loop at the right of the diagram, we have to be able to slide it throuh the strands to a loop on the left, and this is only possible if the knot is actually a knot and not a 2-component link, that is why is is important to recall the condition that for 130 2-essentiality and Families of Knots

T (p, q), p and q have to be relatively prime. Appendix A Topology Miscellanea

Lemma 5.14 (Dehn). Let M be a 3-dimensional manifold and D a disk in M with possible self-intersection and whose boundary C = ∂D is a simple closed embedded curve without singular points of D. Then, there exists another disk D1 which is embedded in M and has the same curve C as boundary.

This important result was ﬁrst stated and proved in 1911 by Dehn. In 1929, 18 years later, a mistake was found in the original proof and the statement remained unjustiﬁed unitl 1957, when Papakyriakopoulos proposed a new proof [49]. He introduced the so- called “tower construction", a process in which one builds the double cover of the image of the disk many times in order to eliminate the singularities. The following is an important theorem about embeddings of 2-spheres, but it has been proved true for arbitrary dimension too, see [10]:

3 Theorem 5.15 (Schoenﬂies). Let S be a smooth embedding of a 2-sphere in R , then 3 there is a homeomophism of R to itself such that the image of S is the unit sphere.

3 Corollary 5.16 (Schoenﬂies). A smooth embedding of a 2-sphere in S bounds a 3-ball on both sides.

See [24] for a proof of these two following standard results of algebraic topology:

Theorem 5.17 (Hurewicz). Let X be a topological space. Then the ﬁrst homology group is isomorphic to the abelianization of the fundamental group:

π1(X) H1(X) ' [π1(X), π1(X)]

131 132 Appendix A

Theorem 5.18 (Van Kampen). Let X be a topological space, such that X = U ∪ V , where U and V are open subspaces of X. Suppose that U ∩ V is path connected and let x0 ∈ U ∩ V . Moreover assume that each such spaces X, U, V and U ∩ V , has a universal covering space. Then, the following diagram, inducted by the inclusions maps at the level of the fundamental groups with x0 as base point, is commutative:

i1∗ π1(U) j1∗ : XXXz π1(U ∩ V ) π1(X) XXz : j i2∗ π1(V ) 2∗

Moreover, π1(X) is the amalgamated product of π1(U) and π1(V ):

π1(X) = π1(U) ∗ π1(V )/N

−1 where N is the normal subgroup generated by the elements of the form i1∗(y)i2∗ (y) for all y ∈ π1(U ∩ V ).

The following is a famous conjecture of Smith from 1939, which became a theorem more than 30 years after its ﬁrst formulation: The case n = 2 was proved by Waldhausen in [63]. The Smith Conjecture was proved in all its generality by several mathematicians, see [40].

3 3 Theorem 5.19 (Smith’s Conjecture). Let f : S → S a non-trivial orientation preserving diffeomorphism of finite order. If its fixed-point set is non-empty, then it is the unknot.

5.2.1 Tori and solid tori

2 We introduce here some basic notions concerning tori and solid tori. Denote by T a 2 1 1 2 1 2 torus T ' S × S and T a solid torus: T' D × S . We have ∂T = T .

Deﬁnition 5.20. A meridian of a solid torus T is a simple closed curve in ∂T that is homologically trivial in the solid torus but not in the torus ∂T . i Let i be the inclusion: ∂T ,→ T , it induces a map at the level of homology groups:

i∗ H1(∂T ) → H1(T )

The meridian generates Ker(i∗) = M. A meridian can also be deﬁned as a simple closed curve whose homology classes are generators of M. 133

Remark 5.21. The concept of meridian is well-deﬁned by the previous deﬁnition up to isotopy and up to orientation. One can show that a meridian necessarily bounds an embedded disk in T .

Deﬁnition 5.22. A longitude of a solid torus T is simple closed curve in ∂T that intersects (transversally) some meridian of T in a single point. Thus a longitude has intersection number +1 or −1 with any meridian.

Remark 5.23. A longitude is deﬁned up to homeomorphisms of T ; it is a generator of

H1(T ), which is isomorphic to Z.

Deﬁnition 5.24. A lens space L is the result of gluing two solid tori by a homeomorphism f of their boundaries.

Remark 5.25. The manifold obtained after gluing is completely determined by the homology class of the image of a meridian m1 of M1, that is by [f(m1)] ∈ H1(∂M2). 2 1 In order to glue a solid torus D × S to another one, we can ﬁrst glue just (a handle) 2 1 D × D and then the remaining 3-disk. To perform the ﬁrst gluing we only need to 2 know the image of the meridian (∂D × {0}). After there is just one way to attach to the boundary a ball.

2 2 1 1 Since for a torus T , π1(T ) = π1(S × S ) = Z × Z is abelian, we have, by Hurewicz’ theorem, H1(∂M2) = π1(∂M2). Therefore, every attaching homeomorphism will be associated with a primitive element of the fundamental group of a torus up to sign, that is by a simple closed curve in the torus. In fact this curve identiﬁes the isotopy class of the image of the meridian m1, a closed curve in ∂M2. 2 Since closed curve in a torus T is identiﬁable up to homotopy by two coprime numbers, by (1, 0) or by (0, 1), a pair ±(p, q) ∈ Z × Z of coprime numbers determined up to sign determines the attaching homeomorphism. The lens space L(p, q) is the lens space obtained by gluing two solid tori along the homeomorphism determined by the pair (p, q). The fundamental group of the Lens Space

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