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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 inter- secting 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 2 N a n-essential secant is strongly essential and moreover it is kn-essential for all k 2 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 essen- zielle 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 .................
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