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Home , Ambient isotopy, Stick number

Geometric Theory

HUI, Wing San

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Mathematics

©The Chinese University of Hong Kong November 2003

The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. 统系i書圓^^ Geometric i Abstract

The space of n-sided (n > 3) polygonal embedded in consists of a smooth in which points correspond to piecewise linear knots (Geometric knots), while paths correspond to isotopies which preserve the geometric structure of these knots. We will discuss the latest development in the study of geometric isotopies versus the topological ones. In particular, for n = 6, the knot space consists offive components, but contains only three topological knot types. These show that the geometric knot equivalence is strictly stronger than topological equivalence. This point is demonstrated by the example of trefoil. In addition, some of the topological features of the lower dimensional knot spaces (3 < n < 6) are described. We will also introduce two other topics, the minimal stick number and the superbridge index. 摘要

嵌入在三維歐氏空間R3的n-邊形紐結空間是一個光滑流形，每一 個點對應於一個分段線性紐結(Geometric knots),而每一條道路則對應 於一個保持紐結的幾何結構的同倫。在這篇論文中，我們會討論幾何

同倫對比於拓撲同倫的一些最新發展。特別當n = 6時，紐結空間有五 個道路連通部份，但是只有三種拓撲紐結類。從此可以看到幾何等價

比拓撲等價嚴格地強。而這一點可以從trefoil的例子看到。另外，我 們也會描述低維紐結空間(3 S n S 6)的一些拓撲特質。我們還會介紹兩 個主題，分SU是 minimal stick number 禾口 superbridge index� Geometric Knot Theory ii ACKNOWLEDGMENTS

I am greatly indebted to my supervisor, Prof. Thomas K. Au, for his continual guidance and constant encouragement and help throughout the period of my postgraduate studies. Contents

1 Introduction 1 1.1 Introduction 1 1.2 Outline of Thesis 2 2 Basic Knowledge of Knot Theory 3 2.1 Preparation 3 2.1.1 Knots, Knot Equivalence and Isotopic Knot . 3 2.1.2 Tame and Wild Knots 5 2.2 Some Invariants and Quantities about Knot 7 2.2.1 Projection of Knot and Crossing Number ... 7 2.2.2 Braids and Braid Index 7 3 Minimal Stick Number 11 3.1 History and Definition 11 3.2 Minimal Stick Number on Some Simple Knots .... 12 3.3 Some Theorems on the Minimal Stick Number .... 14 4 Superbridge Index 22

iii 4.1 Definitions of Bridge Index, Superbridge Index and Total Curvature 22 4.2 Superbridge Index and Braid Index 25 4.3 Relations between Bridge Index, Superbridge Index and Total Curvature 29 4.4 Superbridge Index and Minimal Stick Number .... 36 5 The Geometric Knot Space 37 5.1 Definition of the Geometric Knot Space 37 5.2 Geometric Equivalence and Topological Properties of the Geometric Knot Space, Geo{n) 39 5.3 The Spaces Geo(3), Geo(4) and Geo(5) 40 5.4 of the Space Geo(6) 43 6 Concluding Remarks 52 6.1 Other Results on the Minimal Stick Number 52 6.2 Minimal Stick Number and Superbridge Index of the 54 6.3 Explorations of the Geometric Knot Spaces 56 Bibliography 58 Chapter 1 Introduction

1.1 Introduction For a knot, we always imagine a string with two ends which has some self-crossing. But in the mathematical viewpoint, or more precisely the topological viewpoint, any two strings with two ends are the same since we can always move one of the two ends to remove all the crossings. So, in mathematics, a knot is formed by joining the two ends to the string to form a closed loop. More formally, in mathematical terms, a knot is a embedded circle in E^ or, sometimes, in the one point compactification of E^, which is the 3-dimensional sphere, S^. And for any two knots, if we can deform them from one to another, we say that the two knots are the same. The formal definition will be given in chapter 2. Actually, the study of knots was first motivated by chemistry. In 1880s', knot was one of the imagined model of atoms, which attracted some people to tabu- lating knots to create the table of atoms. Now, we know this model is incorrect, but knot has entered another field of chemistry. In 1980s', about hundred years later, people discovered that the DNA molecules are knotted. Also, people found that the properties of knotted molecule depend on the knot type. More details about this story can be found in Adams's book [1].

1 Geometric Knot Theory 2 1.2 Outline of Thesis In this thesis, we will discuss some new developments in knot theory and three topics are included. The thesis is organized as followings. In chapter 2, we begin our study from the classical knot theory as a prerequisite, which includes the definition of knots and knot equivalence. Then, we will introduce the crossing number and braid index which will appear in the contents. In chapter 3, we discuss our first topic, the minimal stick number, which is about the study of knotted polygons. In our discussion, we will give the definition and will discuss some methods of calculating the minimal stick number. Also, some examples are given. In chapter 4，we study the second topic, the superbridge index, which is motivated by the study of bridge index in classical knot theory. We will define the superbridge and give some relations of the superbridge index to the others, such as the bridge index, the total curvature and the braid index. In chapter 5, we have the geometric knot spaces which are the spaces of polygonal knots. We will define the geometric knot spaces for edge numbers n with n > 3 and study some topological properties of the spaces. Then, we will discuss the cases for n = 3,4,5,6. In chapter 6，we will discuss some results that are related to our topics and some areas for further research. Chapter 2 Basic Knowledge of Knot Theory

In this chapter, we will first discuss some basic knowledge of knot theory. Then we define some knot invariants including both the classical one and the recent one.

2.1 Preparation 2.1.1 Knots, Knot Equivalence and Isotopic Knot In this section, we are going to discuss what a knot is, and give the definition of equivalent knots. First, we define our main object, knots.

Definition 2.1.1. {Definition of Knot) A knot is a subset, K, of R^ such that there exists an of the unit circle, S^, into R^ whose image is K. That is, there exists M^, such that f{S') = K.

In the view of topology, spaces are considered to be the same if they are home- omorphic. But from the above definition, cannot distinguish

3 Geometric Knot Theory 4 any knots, since knots are always homeomorphic to S^. So we need to find an appropriate definition for knot equivalence.

Definition 2.1.2. {Knot Equivalence) Two knots in Ki and K2 are said to be equivalent if there exists a homeo- morphism, h, of onto itself such that h{Ki) = K2,

Clearly, knot equivalence is really an equivalence relation. Under this equiv- alence relation, we have the following definition.

Definition 2.1.3. (Knot Type) The equivalence class of a knot K is said to be the knot type of K and denoted by [K]. That is, for any K' e [K], K' and K are equivalent.

Remark: If a knot is equivalent to the circle x^ = = it is called an or a trivial knot, and the corresponding knot type is called the trivial type.

Besides the above definition of knot equivalence, we have another concept to describe when two knots are considered to be the "same". That is the isotopy type of a knot. As a preparation of the definition of isotopy type of a knot, we need the concept of the ambient isotopy of a topological space.

Definition 2.1.4. (Ambient Isotopy) The ambient isotopy of a topological space X is a continuous mapping H from X X [0,1] to X such that H(.,t) = ht{-) is a from X to X for all t G [0，1] and ho(-) 二 H(� t)is the identity map of X.

Now, we give a formal definition to the isotopy type of a knot. Geometric Knot Theory 5

Definition 2.1.5. {Isotopy Type of Knot) Two knots in Ki and K2 are said to be the same isotopy type (or isotopic) if there exists an ambient isotopy H of such that hi{Ki) = H(Ku 1) = K2. Clearly, if two knots are isotopic, then they are equivalent. On the converse, we have the following remark:

Remark: If two knots Ki and K2 are equivalent, then they are isotopic up to a reflection. More precisely, if the homeomorphism, h, in Definition 2.1.2 is orientation preserving, then they are isotopic. If h is orientation reversing, then Ki and r{K2) are isotopic, where r is any reflection in R^.

2.1.2 Tame and Wild Knots We have defined what a knot is. Actually, in many studies of knot theory, also in our study, only one specific type of knots is considered, which is called tame knots. In fact, in most of the applications, only the tame knots appear. In this section, we will focus on defining this type of knots. Before defining the tame knot, we need to define a special kind of knots first, the polygonal knots. Definition 2.1.6. {Polygonal Knot) A polygonal knot is a knot which is the union of finitely many closed straight line segments. That is a polygon embedded in The line segments and the endpoints of the segments are called edges and vertices respectively.

Notation: For any polygon P in with vertices {vi,v2, • • • we write P =

Remark: Polygonal knots are sometimes called piecewise linear knots.

We can now use the polygonal knot to define our target of this section, the tame knot.

Definition 2.1.7. {Tame Knot) A knot is said to be tame if it is equivalent to a polygonal knot. Those knots are said to be wild if they are not tame.

In fact, it can be shown that if a knot K is rectifiable and is given as the image of a map 7 : [0,/] where 7(5) = (x{s),y(s),z{s)) with arc length s, with first order derivative 7' exists and is continuous for all s, then K is tame. In particular, a C°°-knot is tame. (The details of the proof can be found in the Appendix I of [9].) Actually, there is a more general result for the tameness of knots, and we will mention it later. Prom now on, we will keep our focus on the tame knots, and the word "knot" in the following content will always mean tame knot. 〇• •

Figure 2.1: Unknot, right trefoil and left trefoil. Geometric Knot Theory 7 2.2 Some Invariants and Quantities about Knot The term “" means a quantity assigned to each knot which depends only on the knot type of a knot. In this section, we will discuss some knot invariants and some quantities of knots.

2.2.1 Projection of Knot and Crossing Number By a projection, we means the orthogonal projection of the entire space M^ into a plane in E^ or the radial projection of R^ into a sphere And the projection of a knot, we mean the image of the knot under the projection.

Definition 2.2.1. {Crossing Number) For each knot K, the crossing number, c[K], of K is defined to be the minimum number of crossings which occur in all projections of all the knots belonging to the knot type [K] into a plane or sphere.

2.2.2 Braids and Braid Index In this section, we will define a knot invariant, the braid index. We first discuss what a braid is.

Definition 2.2.2. {Braid) Let be the cube {(a:, y, z)\0 < x,y,z< 1} in R3’ and let n be a positive integer. Let Pi =(击’ i, 1) and Qi 二（；^’ 0)，i 二 1，...，n which lies on the top and bottom of P respectively. Let Si’... ’ be n mutually disjoint arcs such that d{si U …U = {尸1，... ’ Pn, Qi，…’ Qn} and for any i, Si intersects the plane {z = c} exactly once, where 0 < c < 1 is a constant. Then, we call b = SiU- • -USn an n-string braid. Geometric Knot Theory 8 I Figure 2.2: An example of braid.

Remark : We say that two braids bo and 6i are ambient isotopic if there exists an ambient isotopy / : /3 X [0,1] 4 such that /山j3 = id, /�id= and /i(6o) = h.

Actually, when we use braids to study knots (or more generally, links^), closed braid is one of the main tools.

Definition 2.2.3. {Closed Braid) Let 6 be a braid with notations in Definition 2.2.2. A closed braid, 6, is formed by connecting /^j to Qi, i 二 1’. •. ’ n, with trivial arcs. That is, the image of the orthogonal projection, p, from b - P to the x^^-plane has no crossing.

y

Figure 2.3: An example of closed braid.

1A is an embedded image of finitely many mutually disjoint unit circle Geometric Knot Theory 9

A well-known theorem, which is proved by Alexander in 1923, said that every knot (or link) can be represented by a closed braid (details can be found in chapter 2 of [4]), so we can always use the closed braids to study the knots. This motivates us to define the braid index.

Definition 2.2.4. {Braid Index) The braid index of a knot, K, is the minimum number of braid strings needed to form a closed braid that is ambient isotopic to K. And we use (5[K] to denote the braid index of K.

Since every knot has a closed braid presentation, the braid index is well- defined. Also, from the above definition, it is clear that braid index is a knot invariant. Now, consider a n-string braid, if n is even, that is n = 2m，then there is another way to close a braid into a knot or link, which is called plat.

Definition 2.2.5. {Plat) For a 2m-string braid, we close it by connecting P2i-i to Pii and Q2i-i to Q2i by 2m simple arcs, i = 1，... ’ n. Then, the resulting knot or link is called a plat. i Figure 2.4: An example of plat. Geometric Knot Theory 10

Obviously, by direct construction, we can always find an isotopy from a n- closed braid to a 2n-plat. So, by the theorem of Alexander, every knot (or link), has a plat presentation. Chapter 3 Minimal Stick Number

In this chapter, we will define a knot invariant, minimal stick number. Then we will state and prove some theorems about the formulation of the minimal stick number.

3.1 History and Definition The study of minimal stick number was first introduced by Randell in his paper [22] in 1994. When knots are used to model molecules such as DNA and other polymers, it is natural to view the knots as polygons, which are the polygonal knots (Definition 2.1.6). In the study of polygonal knots, it is natural to ask a question, that is, given a particular knot type, how many edges are needed to visualize the knot as a polygonal knot? It is clear that once we know that a knot can be formed by n edges, then for any N >n,we know the knot can be formed by N edges. This motivates us to find the least number of edges which is needed to form a specific knot. Towards this goal, we have the following definition.

Definition 3.1.1. {Minimal Stick Number) The minimal stick number, s[K], of a knot type [K] is defined to be the smallest number of edges required to realize [K] as a polygon. 11 Geometric Knot Theory 12

Prom the definition of tame knot (Definition 2.1.7), the knot type of a knot (tame knot) must include a member of polygonal knots. So, given any knot, K, we can always find a polygonal knot which is equivalent to K. So the minimal stick number is a well-defined knot invariant.

3.2 Minimal Stick Number on Some Simple Knots In this section, we will calculate the minimal stick number of the unknot and the . Obviously, in order to form a closed loop in at least three edges are needed. So, for the unknot, the minimal stick is 3 (see Figure 3.1), that is s [unknot] = 3. (3.1)

Figure 3.1: A polygonal exampl e of unknot. The next example is the trefoil knot. In Figure 3.2, we show the polygonal examples of both left trefoil and right trefoil. Prom the definition of minimal stick number and these examples, we know that s[trefoil] < 6. (3.2) In fact, the above inequality is an equality, that is, s[trefoil] = 6. (3.3) Geometric Knot Theory 13

Figure 3.2: A polygonal example of left trefoil knot and right trefoil knot.

Actually, we will show a stronger result, that is, any polygonal knot with n edges, 3 < n < 5, must be equivalent to the unknot. And this will be an direct consequence of the theorems which will be discussed in the next section. The third example is the figure-eight knot. In Figure 3.3, the polygonal example with seven edges is shown. So, we have s[figure-eight] < 7. (3.4)

Figure 3.3: A polygonal example of figure-eight knot.

Again, the inequality is in fact an equality, that is, s [figure-eight] = 7. (3.5) And this fact is also an direct consequence of the theorems which will be discussed in the next section. Geometric Knot Theory 14 3.3 Some Theorems on the Minimal Stick Num- ber In this section, we will state and prove some theorems on the minimal stick number. These will give us some methods to estimate or calculate the minimal stick number of some specific knot types. Firstly, we discuss the relation between the minimal stick number and the crossing number defined in Definition 2.2.1. Prom Negami's paper [21], we have the following theorem,

Theorem 3.3.1. (Theorem 7 in [21]) For any knot K, if K is not the unknot, then we have the following inequality, 5 + (36)

Proof. Lower hound: For any n-sided polygonal knot K in we can choose an edge eo and project K into a plane which is perpendicular to cq. Moreover, by moving a little if needed, we can assume that eo is the only edge of K which is perpendicular to the plane. Now, let the projection be p : —> so p{K) is a (n - l)-sided polygon. For each edge of p(K), there are at most (n-4) crossings, so p{K) contains at most l)(n- 4) crossings. By the definition of crossing number (Definition 2.2.1), we have c[K]<^in-l)(n-4) n^ - 5n + 4 - 2c[K] > 0 ^ 5 + ^25 - 4(4 - 2c[K]) n>2 、5 + ^/9 +嘲） n > - •

Now, since the above equation is true for any n-sided polygonal knot K, it is also Geometric Knot Theory 15

true for n = s[Ar). This give us the lower bound of (3.6), that is,

他 5 + (3.7)

Upper bound: Let G be the 4-regular graph formed by a minimum crossing projection of [K] (since K + unknot, G is always defined). Note that, by our assumption, the following graphs (Figure 3.4) will not occur as a part of G. X) D

Figure 3.4: Diagrams that will not appear in the G

Let G' be the graph obtained from G by replacing each pair of multiple edges with a single edge. So G' is a simple graph. Now, recall a well-known theorem in graph theory: Every planar simple graph is homeomorphic to a planar simple graph with each edge a straight line. That is there exists G" (with linear edges) such that G" is ambient isotopic to • •Q V K G G' G" Figure 3.5: Example on the deformation. Geometric Knot Theory 16

Now, we split each vertex of G" into two points in (upper and lower) and join them by straight lines corresponding to edges of G. After these processes, we form a polygon Q in R^ which is projected to G". Note that, Q may have some self-intersections and each of them lies on two edges contained in one vertical plane and is projected to an edge in G" which corresponds to a pair of multiple edges in G. Finally, by pushing off those edges to eliminate the crossing and by taking care of this process, we will get a new polygon Q' in M^ which is a polygonal representative of K. Prom the constructions, we have e(Q') = e(Q) = e(G), where e(-) is the number of edges. And from the definition of G, we have viG)=c[K], where ？;(•) is the number of vertices. Now, since G is a 4-regular graph, we have 2e(G) = HG). Hence, we get e{Q') = 2c[K]. From the definition of minimal stick number, we have the following, s阅 < e(Q') => s[K] < 2clK], which is the upper bound of (3.6). • Now, by Theorem 3.3.1 we have the following corollaries.

Corollary 3.3.2. A polygonal knot, K, in R^ with 3 to 5 edges must be the unknot. Geometric Knot Theory 17

Proof. Assume K is not the unknot, from the definition of minimal stick number, we have s[K] < 5 in any given case. Applying Theorem 3.3.1, we have s[K] < 5 5 + v/9 + 8c[i^]) ^ ^ 2 9 + 8c[K] < 25 玲 c[K] < 2 c[K] < 3. Now, it is a well-known fact that the unknot is the only knot that has crossing number less than 3, which is a contradiction. So, K must be the unknot. •

Corollary 3.3.3. The minimal stick number of the trefoil knot is six, that is the equation (3.3).

Proof. Prom (3.2), we known that the minimal stick number of the trefoil knot is at most six. Now, from the corollary 3.3.2，we know that the trefoil knot cannot be obtained by edges less than six. That is, the minimal stick number is at least six. So we get the equation (3.3). Alternatively, since it is known that the crossing number of the trefoil knot is 3, by Theorem 3.3.1, for the lower bound, we have s [trefoil."1]、 > 5 ——+ ^ ^[trefoil] > > 5 玲 s[trefoil] > 6. For the upper bound, we have s [trefoil] < 2c[trefoil] 今 s[trefoil] < 6. Geometric Knot Theory 18 which is given by (3.2). So, we get (3.3) again. • We have just calculated the minimal stick number of the trefoil knot by using the lower bound (3.7) obtained from Theorem 3.3.1. However, when applying the equation to the figure-eight knot, we can only get s[figure-eight] > 6 again, which cannot reach our goal, s[figure-eight] = 7 (equation (3.5)). So, in order to solve this, we need another lower bound, and we have the following theorem,

Theorem 3.3.4. (Theorem 4 in [5]) For any knot K, if K is not the unknot, then we have the following inequality,

,[；(3.8) Proof. Suppose K is an polygonal knot with n edges. We can relabel the vertices of K, so that vi is a point on the boundary of the convex hull of K. So, we can find a plane H such that 丑 n = {t/i} and K lies entirely on one side of H. Now, let be a sphere centered at v\ such that K is inside S. Let r : K - {？;}i ^ S be the radial projection. Since K lies inside the region bounded by a hemisphere of S and H, so r(K) lies entirely in a hemisphere of S cut by the equator n i/. Note that r{viv2 - {？;i}) = r{v2) and r(VnVi - {t;i}) = r(Vn). By moving a little if needed, we can assume r[K - = F consists of n - 2 great circular arcs on S. Suppose r has c(r) crossings. Since F is in a single hemisphere of 5, each pair of arcs will intersect at most once. Now, we count the maximum number of crossings that can occur . For r(fa 1*4),..., r(?;„_2i'n-i), each can intersect with at most n - 5 arcs. For and r(vn-iVn), each can intersect with at most n - 4 arcs. So we have e(r)S>-4)(n — 5) + 2(n-4)l=(�-3)2(n-4).

Let £ > 0 be small enough such that B^ (e-ball centered at Vi) intersects K Geometric Knot Theory 19 in two segments of the edges V1V2 and VnVi only. Suppose naBe = {gi},

iJnaB,nP(t;i，^，t;3) = te}， where P{vi,v2, vs) is the half-plane containing V2 and bounded by the line segment V1V3 (this is well-defined because Vi,V2,V3 are not collinear). We can deform viQi to Viq2 U ai, where ai is the great circular arc from 仍 to qi. Since ai lies on the same plane as V2V3 which contains vi, r(ai) U r(v2V3) forms a great circular arc on S from r{q2) to r(v3). Since the possible crossings on this arc have already been counted, after this deformation, the upper bound on the total number of crossings still holds. Similarly, suppose ^nVi n dBe = {93}, where P(vn-i,vn,vi) is the half-plane containing Vn and bounded by the line segment V1V3 (this is also well-defined). We deform q^vi to a2 U 如Vi, where ；0 2 is the great circular arc from qs to 彻.Since 0:2 lies on the same plane as Vn-iVn which contains Vi, r{vn-iVn)^r{a2) forms a great circular arc on S from ”(?;打)-1 to rfe). Again, since the possible crossings on this arc have already been counted, after this deformation, the upper bound on the total number of crossings still holds. Finally, we deform q4i>i and v^ by moving Vi into the sector q4Viq2 until they coincide with an arc, 0:3 along the equator dBeOH. Obviously, this transformation does not change the knot type of K. And the projection, r', of the new knot is just the n - 2 arcs of r with two ends extended by r(Q:i) and r(Q;2)，together with a arc r{az) on the equator n 丑 which joins the end points r(g2) and r(q^). Since T is outside the equator, F' has the same number of crossings as F. Hence Geometric Knot Theory 20 we get

n2 — 7n + 12 - 2c[K] > 0 \ 7+v/l +刚 ^ — 2 . Clearly, if we assume n is equal to s[K], the proof still works. Thus, we get which is the equation (3.8). 口

Remark : In fact, the inequality (3.8) is really an improvement of the inequality (3.7). It is because, x/1 + SclK] > 1 4 + 47l + 8c[i^] >8 4 + 4v^l + Sc[K] + (1 + 8c[K]) >9 + 8c[K] (2 + + > (v^9 + 8C_2 2 + + 8刺 > ^9 + SclK] 7+x/l + 8c阅 \ 5 + x/9 + 8刺 2 - 2 • Now, we can use this theorem to calculate the minimal stick number of the figure-eight knot.

Corollary 3.3.5. The minimal stick number of the figure-eight knot is seven, that is the equation (3.5).

Proof. Prom (3.4), we have known that the minimal stick number of figure-eight knot is at most seven. Now, since it is known that the crossing number of the Geometric Knot Theory 21 figure-eight knot is 4, by Theorem 3.3.4, we have

s[figure-eight] > 了十严 > 6 s [figure-eight] > 7. Hence, we get the equation (3.5). • Combining the two theorems (Theorem 3.3.1 and Theorem 3.3.4), we have the following,

Theorem 3.3.6. For any knot K, if K is not the unknot, then we have the following inequality, 7+yi^ 娜 2稱 (3.9) Chapter 4 Superbridge Index

The main target of this chapter is a knot invariant, the superbridge index. We will give the definition of the superbridge index and discuss some relations between it and other knot invariants, including the minimal stick number.

4.1 Definitions of Bridge Index, Superbridge In- dex and Total Curvature In this section, we will discuss two knot invariants, bridge index and superbridge index and one quantity, total curvature. The definitions we use here are based on [19], [12] and [11]. We first define the term crookedness of a knot, which is closely related with our three targets.

Definition 4.1.1. (Crookedness) Given any knot K, we parametrize K by the vector function �(i)， e tS^. For any unit vector v in we define the crookedness of K with respect to v as the number of local maxima of the function v •��and w’ e denote it by h^(K).

22 Geometric Knot Theory 23

Remark: The set of all unit vectors in R^ is actually the unit sphere. And the notation is always reserved for the unit sphere in Next, we will define the and the and the bridge index of a knot (which is called crookedness of a knot in Milnor's paper [19]),

Definition 4.1.2. (Bridge Index) For any knot K, we define the bridge number, b(K), of K to be the minimum of its crookedness with respect to all directions. That is,

h(K) = min b芬[K). 11-11=1 �, Then, we define the bridge index, b[K] of a knot, K, to be the minimal bridge number over all knots that belong to its knot type [K]. That is, blK] = min b(K') = min min bJK'). L J K'e[K] ^ ‘ K>e[K\ 11^711=1 ‘ Actually, this is not the original definition of the bridge index. When it was first introduced by Schubert in 1954, the definition is based on the diagram representation of a knot. When a knot is tame, we can show that these two definitions give the same number. Also, in [19], Milnor proved that a knot is tame if and only if the bridge index of the knot is finite. We then define two similar objects, the superbridge number and the super- bridge index of a knot, which are first studied by Kuiper [12] in 1987.

Definition 4.1.3. (Superbridge Index) For any knot K, we define the superbridge number, sb{K), of K to be the maxi- mum of its crookedness with respect to all directions. That is, sb{K) = max b^{K). �IMI=i) �) Geometric Knot Theory 24

Then, we define the superbridge index, sb[K] of a knot, K, to be the minimal bridge number over all knots that belong to its knot type [K]. That is, sftfiiT] = min sb(K') = min max bJK'). K'e[K] K'£[K]丨问1=1 Prom the above definitions (Definition 4.1.2 and Definition 4.1.3), it is clear that the bridge index and the superbridge index are knot invariants, as they are given by taking minimal over the knot types. The last one we will define is the total curvature of a knot. Note that, our definition is deferent from the one in Kuiper's paper [12] by multiplying tt. Definition 4.1.4. (Total Curvature) For any knot K, we define the total curvature, k{K), of K to be 2冗 times the average value of h^{K) over all unit vectors v. That is where dA is the surface element on

Actually, this definition does not only apply for a knot, but also apply for any closed curve in M^. In fact, it can be shown that this definition is agree with the other definitions, (1) If P is a closed oriented n-gon, then n = (4.1)

where a^, i = 1,..., n are angles between successive edges (considering the edges as vectors). (2) If C is any closed curve, then = sup /c(P), Per Geometric Knot Theory 25

where V is the set of all polygons inscribed in C and k(P) is defined as (4.1). In fact, this is the original definition in Milnor's classical paper [19]. (3) If C is a closed curve of class C^ parameterized by arclength s, then

where p is the parametrization. Note that, we will use these facts without proving them, and the proof can be found in [19].

4.2 Superbridge Index and Braid Index In this section, we will discuss the relations between the superbridge index and braid index. We first look at the braid index.

Theorem 4.2.1. {Theorem Cin [12]) For every knot K, we have the following inequality, sb[K] < 2(3[Ki (4.2) where P[K] is the braid index of K.

In order to prove this theorem, we need the following lemma.

Lemma 4.2.2. The superbridge number of the curve r]: {x, y, z) = (cos u, sin u, cos^ u), u mod 27r, is two. That is 56(77) = 2. Proof. After projecting 7] to the direction v = {vi,v2,v3) (a unit vector), we get (f)�=v) vi cos n +1;2 sin w + v^ cos^ u. Geometric Knot Theory 26

In order to find the number of maxima, we first differentiate (j) with respect to u and find zeros. So, we have the following equation,

^du = -•Ui sinU + V2C0SU- 2vs cosusmu = 0. (4.3) In order to find the number of solutions of (4.3), we have the following cases. (1) For V2 = 0. We have

-Vi sin u — 2vz cos u sin u = 0 sin u{vi + cos u) = 0 => sin u = 0 or Vi + 2v3 cos u = 0. So the solutions are w = 0, tt and at most two more from Vi + 2v3 cosu = 0. (2) For V2^0 and w = tt is not a solution for (4.3). We can set . 2s � l-s2 謹=1：；：^ and cosu = Y^ to find all the solutions. So we get -倘+腦--•盼。 -t;i(2s)(l + s2) + V2{1 - + s2) — 2^3(1 " s'^){2s) = 0 + {2vi - 4t;3)s3 + {-2vi - -V2 = 0, which has at most four real solutions in s. So, there are at most four solutions for (4.3). (3) For and u = tt is a solution for (4.3). We have

—Dl sin TT + cos TT - 2^；3 sin TT cos TT 二 0 V2 = 0, which is a contradiction. Geometric Knot Theory 27

Hence, the number of solutions of (4.3) is at most four. Then, since r; is a closed curve, the number of maximum and the number of minimum are equal, and their sum must be less than the number of solutions of (4.3). So there are at most two maximum, that is sb(r]) < 2. Now, we consider (1*1,1*2, vs) = (0,0,1), we have two maximum at li == 0 and u = tt, so we get 56(77) = 2. • Now, we prove the Theorem 4.2.1. Proof. (Theorem 4.2.1) Since we have already known that every knot has a closed braid representation, we can represent the knot type of K by closed braid as following. Let a r-braid knot (ambient isotopic to i^)with r 二 be given for small e > 0 by Ke : (cosrt(l + £:Ai(t)),sinrt(l+£Ai(()),aDs2ri + £A2(t))， where Aj + A2 < 1 and Ai and A2 are periodic in t mod 27r. So, for small £ > 0, Ks lies inside a solid torus with vertical disc of radius e. Now, since Ai and A2 are periodic and they can be expanded into Fourier series and the convergence is uniform. Then, we can approximate Ai and A2 by /ii and //2，where /ii and 112 are finite linear expressions in cos nft and sin rijt, rij G N, j G / and / is a finite set. Both of and /i2 are the polynomials in cost and sint. Moreover, we can always assume N 二 maxrij- > 2r. Hence, by replacing Aj by //《，i 二 1,2, we can replace K^ by an isotopic knot K:, K'e : (cos7-f(l + £/^i(t)),sinr^f(l + £/ii(t))’ms2rt + £/i2(t)). Next, we project K'^ to a fixed direction v = {vi,V2, vs) (a unit vector), we get

(peiu) = Vi COSrt{l - efii{t)) +V2 sinrt(l - e/xiW) + V3(cos2 rt + �). Then, we can rewrite as a polynomial of cost and sint, and collect terms in e as (l>e{u) = Pf(cos t, sin t) + eP^^^(cos t, sin t), Geometric Knot Theory 28

where Pp”） and P^n� arepolynomials in cost and sin^ of degree 2r and N re- spectively. In order to find the number of maxima, we first differentiate with respect to t and find zeros. So, we have the following equation, * = Q(产)(cos t, sin t) + eQ广)(cos t, sin t) 二 0’ (4.4) where and are polynomials in cos 亡 and sin 亡 of degree 2r and N re- spectively. Consider e = 0, K'q is a. r-fold covering of the curve 7) in Lemma 4.2.2. So, by Lemma 4.2.2’ Qf^\cost,sint) = 0 has at most 4r solutions in t. Also, by translating t with a constant if needed, we can assume t = tt is not a solution. Now, let Qf^r)(cos7r’sin7r) = Ci ^ 0 and 广)(cos7r，sinTr) = C2, we can always choose �suc0 h that Ci + e'C) 0, for all e < e'. It means that t =-k is not a solution for (4.4) for all e < e'. From now on, we assume £ < e' as above. Again, since t = tt is not a solution, we can set • , 2s J , ； sin t = 1 + s2 and cos t = 1 + s72 to find all the solutions. So we get + + eB_(s) = 0 (4.5) where 力 and B则� are polynomials in s of degree 4r and 2N respectively. For £ = 0, there are N — 2r roots of i, N — 2r roots of -i. So, by continuity, there exists a small Sv > Q such that for all e < e^, there are N - 2r roots in a small neighborhood of i and N - 2r roots in a small neighborhood -i, where the intersection of real line and each neighborhood is empty. So, there are at most 4r real solutions for (4.5). This means in the direction v we have at most 2r maxima. That is, for any e < 仔£ ， h,{K) < 2r. (4.6) Geometric Knot Theory 29

Now, since the choice of e^ depends continuously on the choice of v and the set of all V is S^, which is a compact set, we can find £ = £o > 0 such that for any V e S^, we still have (4.6). So we have sblK] < sb{K'J <2r = 2/?阅’ which is the equation (4.2). •

4.3 Relations between Bridge Index, Superbridge Index and Total Curvature Clearly, by the definitions in the section 4.1, for any knot K, we have the following trivial facts, b{K) < < sb{K) blK] < sb[K] In this section, we will discuss some deeper results in the relations between the three quantities. Some of them can be found in [10, 11，12]. First, we want to show the effect of linear transformation on the bridge number and superbridge number of a knot. Theorem 4.3.1. Given any knot K and any nonsingular linear transformation g G GL(3,M), we have b{g{K)) = b{K) sb{g{K)) = sb(K). In particular, we have b[g{K)] = blK] sblg{K)] = sb[K]. In order to prove the theorem, we need the following lemmas, Geometric Knot Theory 30

Lemma 4.3.2. For any unit vector v e M.^ and any nonsingular linear transformation g G GI/(3,M), we define 伊 to be the unique unit vector in (乂护))丄 satisfying g{v) • v^ > 0. Then the mapping bijection from S^ to S^. Proof. First, we show that 伊 is well-defined for all g G GL(3,M). Since the dimension of 丄))丄 is one, the intersection between 丄丄)) and has only two points with position vectors Vq and -vq. Also, g{v) -vq^Q, we prove this by contradiction. Suppose not, we have 9{v).询二 0 9{v) g{v^) =>• V e v^, contradiction. So, there is one and only one of Hq and -vq which satisfies our condition. Next we want to show the mapping v y-^ iP is & bijection from S^ to In order to prove this, we need an explicit definition of v^. Consider iT丄=span(ui,^2), where Ui and U2 are unit vectors such that Ui x U2 = v. Then we have = spa.n{g{ui),g(u2)). This means v^ = span(i^(ui) x ff(u2))- Note that, if g is orientation preserving, that is det(g) > 0，then g{v). ig{ui) x g{u2)) > 0. If g is orientation reversing, that is det(^) < 0, then g{v) . {g(ui) x ^(^2)) < 0. So we

"e’ 〜陶)鑑X纖 （4.7) For i = 1,2, we have the following, -1 ( gjui) \ = 1 g VIIpWII； — II^WII “

So, put into (4.7) and use the fact sign(det(p-i)) = sign(det(p)), Geometric Knot Theory 31 we get

For injectivity, consider the following 对=谓 冷 W)厂1 =间)厂1 =>• Vi = V2. For surjectivity, for any w G S^, we have e and {w^ = w. So, the mapping is really a bijection. •

By using 逆,which is defined in Lemma 4.3.2, we have the following lemma,

Lemma 4.3.3. For any unit vector v e and any nonsingular linear transformation g e GL(2,R), the equality K乂 gm = HK) holds for any knot K.

Proof. At each local maximum point m of the projection p oi K into the line Rv, let Pm be the level plane at m with respect to p, which is clearly perpendicular to V and touch K eX m locally. Then g{Pm) is perpendicular to v^ and touch g{K) at g{m) locally. By the definition of v^, g{m) is a local maximum point of the projection of g{K) into the line RiP. So, we have bv{K) < b“g(K)). From the proof of Lemma 4.3.2, we have (伊)厂工=v, so, we get

bv^igiK)) < b(仰乂g(K)) = K

Now, we can prove the Theorem 4.3.1. Proof. (Theorem 4.3.1) By Lemma 4.3.2, the mapping of i; i)^ is a bijection, and by Lemma 4.3.3, bij(K) = bi^9{g{K)), so we have b{K) = g 皆 (幻== = b(g(K)), sb(K) = = m^bgy{g(K)) = 沪(g(K)) 二 sb(g(Ky). Now, by taking minimum over the knot type, we finish the remained parts. • Next, we will discuss how the total curvature is related to the bridge number and the superbridge number of a knot. Firstly, we have the following lemma,

Lemma 4.3.4. For any knot K and any unit vector v such that b{K) < oo , there exists a family of nonsingular linear transformations {g^ G GL(3,R)|0 < A; < 1}, such that,

k-*0lim^feW ) = 27rh{K). Proof. Firstly, we choose a coordinate system, {xi,x2,xs), for R^ such that the xi axis is parallel to v. For any 0 < A; < 1, we define gk G GL(3,R) by = {xi, kx2^ kxs). Since bg{K) < oo, it is clear that as /c 0, QkiK) —> Pq, where Pq is a degenerated polygon, and each maximum point and each minimum point of K will respectively correspond to a maximum point and a minimum point of Pq which are also vertices, that is bg{Po) = Now, we calculate the curvature of Pq, = 7r(6,-(Po) + M^o)) = 27r6,^(Po). So, we have \im K{gk{K)) = «(Po) = 27r6,-(Po) = 一 A: 0 • Geometric Knot Theory 33

Next, we have the following theorem,

Theorem 4.3.5. For any knot K with b{K) < oo, we have the following equation,

“2f3，严))二稱 （4.8) And if sb{K) < oo, we have the following equation, sup K{g{K)) = 27rsb(K). (4.9) 9eGL{3,R) In particular, for any knot K, we have 蟲广⑷=2寧1. (4.10)

Also, for any knot type [K], there exists a knot K G [K], such that for any g e GL(3,R), we have 27rb[K] < K{g{K)) < 27TsblK]. (4.11)

Proof. For any knot K with sh(K) < oo and any g e GZ/(3,M), by definition, we have 滅mV) < '^{gm < 27Tsb{g{K)). By Theorem 4.3.1, we get 27Tb{K) < K(g(K)) < 27rsb{K). Now, by considering Vi G with &贞{K) = b(K) and Lemma 4.3.4 we have a family of nonsingular linear transformations {gk G GZ/(3，R)|0 < A; < 1} such that limKte(i^)) = 27rb^,(K) = 2'Kb{K). This implies the equation (4.8). Similarly, by considering V2 G S^ with bg^(K)= sb{K), we get the equation (4.9). Geometric Knot Theory 34

Now, for any knot K, since K is tame, there exists a polygonal knot P e [K]. Clearly, h[K\ < h{P) < oo. Then, we can choose Kq G [K] such that B(KO) = b[K] < oo. By (4.8), we get

^mf^, < 姊结i ’购 <9iKo)) = = 27r6[i^]. However, by definition, we have so, we get (4.10). For (4.11), the lower bound is trivial, so, we only need to prove the upper bound. For any knot type [K], there exists a polygonal knot P € [K]. So, we have sb[K] < sb(P) < oo. Similarly, we can choose Kj^ G [K] such that sbiJC'Q) = sblK] < oo, then by (4.9), we get f^igiK)) < sup K{g{K'o)) = 27rsb(K'o) = 27rsb[K]. FF€GL(3,M) This finishes our proof. • Clearly, the upper bound of equation (4.11) cannot be smaller. For the lower bound, actually, Milnor [19] has shown that the equality does not hold for any knot and clearly, it cannot be larger. Finally, we want to show that the bridge index gives an upper bound to the superbridge index.

Theorem 4.3.6. For any knot K, the following inequality holds, sb[K] < 5b[K] + 3. Geometric Knot Theory 35

Proof. Let h[K] = n. The proof is based on the fact that any n-bridge knot has a 2n-plat presentation and the 2n-string braid involved can be chosen to be a 272 - 1-string braid with a single straight string, say the 2n-string. Now, we deform the 2n-plat (Figure 4.1), K', and form a singular closed braid, S, by replacing the arcs Ai-i尸2i and Q2i-iQ2i with two strings P2i-iQ2i and Q2i-iP2i, where the two strings cross once, i = 1,... ,n - 1. (nJ ( rvri

Figure 4.1: The deformation of the braid. Then we afollow the proo f of Theore^m 4.2.1, we have ^ 2sb{S) < 4(2n - 1) = 8n - 4. Now, we recover K' by removing the n - 1 singular points, and this process will give totally almost 2n-2 more maximum or minimum points for every direction V e So, we have 2sbiK') < 2sb(S) + 2n - 2 < lOn - 6. Finally, we get sb[K] < sb(K') < 5n - 3 = 5b[K] - 3. • Geometric Knot Theory 36 4.4 Superbridge Index and Minimal Stick Num- ber In this section, we will relate the two knot invariants, the minimal stick number and the superbridge index.

Theorem 4.4.1. {Proposition 2.5 in [11]) For any knot K, we have the following inequality, 2sblK] < s阅.

Proof. Firstly, we choose a polygonal knot P such that P G [K] and the number of edges of P is equal to s[K]. Now, for any v G S^, any maximum with respect to V must appear at a vertex or a whole edge, so we have the following, (P)+ Then by taking maximum over v, we get

2sb[K] < 2sb{P) = 2MAX6^-(P) < slK]. • Prom the above, the superbridge index gives a lower bound to the minimal stick number. However, given any fixed bridge index, the knot with the bridge in- dex can have a arbitrarily large crossing number, by Theorem 3.3.6, the minimal stick number is therefore unbounded, but, by Theorem 4.3.6, the superbridge index is bounded, so this lower bound is not accurate. Chapter 5 The Geometric Knot Space

In this chapter, we will study the geometric knots. We will give the definition of geometric knot space, then we will discuss some properties and details of the geometric knot space. Here, we will follow the main idea which is given by Calvo in [61.

5.1 Definition of the Geometric Knot Space The idea of studying the geometric knots was first introduced by Randell [23’ 24]. In this section, we will give the definition of geometric knot space and some properties on the geometric knot space. Actually, for a fixed n, the n-sided geometric knot is just a n-sided polygon. And the geometric knot space is just the set of all n-sided embedded polygons in

For any polygon P =< ‘ • • ,VN >, which may be non-embedded, we call the vertex Vi to be a root and we choose a orientation for P according to the labelling of the vertices. Now, we can view P as a point of by listing the coordinates of the vertices starting from Vi and proceeding in sequence which are determined by the orientation. With this viewpoint, the origin of 股打3 is

37 Geometric Knot Theory 38 corresponding to the degenerated polygon in R^ which is in fact the origin of R^. Now, we first define the set of points in corresponding to all non-embedded polygons, and we will call the set discriminant After that, we can use the discriminant to define the geometric knot space. For the formal definition, we first consider a polygon P =< •^1,1*2’ …,Vn > such that ViVi+i intersects VjVj+i with i ^ j, i ^ j + 1 and i + 1 — j, i,j are mod n indices. Then, we have - Vi) X {Vj — Vi) . {Vj+i - Vi) = 0, (Vi+i - Vi) X {Vj 一 Vi) • — V^ X {Vj+i - Vi) < 0, (巧+1 — ^j) X — Vj) • {vj+i - Vj) X (^^i+i - Vj) < 0. Clearly, any polygon satisfies the above conditions must be non-embedded (for any ij with + l and i + 1 # j). And for any fixed since each equation is, in fact, formed by a polynomial of the coordinates of the above conditions define a semi-algebraic variety in E^". There are totally -n(n - 3) such semi-algebraic varieties, and any point corresponding to a non-embedded polygon must lie in the closure of one of these semi-algebraic varieties. So, we have the following, Definition 5.1.1. (Discriminant and Geometric Knot Space) We define the discriminant, E(n), to be the union of the closure of each of the in(2 n — 3) semi-algebraic varieties. And we define the geometric knot space, Geo(n), by Geo{n) = -E(n). The space Geo{n) is also called the embedding space of rooted oriented n-sided geometric knots. Geometric Knot Theory 39 5.2 Geometric Equivalence and Topological Prop- erties of the Geometric Knot Space, Geo(n) In this section, we study the topological properties of the geometric knot spaces, Geo(n). We first have the following theorem,

Theorem 5.2.1. For any n, Geo(n) is a 3n dimensional manifold.

Proof. This is just a trivial consequence followed from the definition. Since E(n) is a finite union of closed subsets of it is closed. So, being a complement of a closed subset, Geo(n) is open. And clearly, Geo(n) is non-empty. So, being a non-empty open subset of E^", it is also a 3n-manifold. • Prom now on, a polygonal knot P G Geo(n) means that the point, p G corresponding to P belongs to Geo{n) Next, we discuss the geometric equivalence. We first consider a path h : [0,1] —)• Geo{n). By the definition of Geo(n), h defines an ambient isotopy from the polygonal knot h{0) G Geo{n) to the polygonal knot h(l) E Geo(n). This motivates us to have the following definition,

Definition 5.2.2. Let Pi and 尸2 be any n-sided polygonal knots, that is Pi,P2 G Geo{n). Then F\ and P2 are said to be geometrically equivalent if they belong to the same path- component of Geo{n). And each path-component is called a geometric knot type. Also, a n-polygonal knot P is said to be geometrically unknot if it is geometrically equivalent to a planar polygon. It is well-defined since all n-sided planar polygons are geometrically equivalent (proved in [2]).

Remark: Geometric Knot Theory 40

The geometric equivalence of two knots implies their isotopic equivalence. How- ever, some examples show that the converse is not true and we will discuss it later.

Now, it is natural to ask a question, that is, for a fixed n, how many geometric knot types (path-components) are there in the space Geo{n)7 For the answer of this question, we have the following theorem.

Theorem 5.2.3. For each n, there are only finitely geometric knot types in Geo{n).

Proof. The proof is not in details and is based on the following facts in real algebraic geometry (can be found in [3]). 1. Closure of semi-algebraic sets is also semi-algebraic. 2. Finite union of semi-algebraic sets is also semi-algebraic. 3. Complement of semi-algebraic sets is also semi-algebraic. From 1 and 2, E(n) is semi-algebraic. Then, from 3, Geo{n) is semi-algebraic. Now, by a theorem in real algebraic geometry, Geo{n) has a finite number of path-components. •

5.3 The Spaces Geo(3), Geo(4) and Geo(5) In this section, we will study some examples of Geo{n) for 3 < n < 5. Actually, in chapter 3, when the number of edges is smaller than 5，we know that only the topological unknot can be formed. So, we know that these three spaces contain only the topological unknot. However, we will show a stronger result, that is the three spaces contain only the geometric unknot. We first consider the space Geo{S). Geometric Knot Theory 41

Theorem 5.3.1. Geo{3) is path-connected.

Proof. Clearly, all polygonal knots in Geo{3) are triangles which are always pla- nar, so Geo{3) is path-connected. • Next, we consider the space Geo(4).

Theorem 5.3.2. Geo{4) is path-connected.

Proof. For the space Geo(4), it consists of the quadrilaterals, and each quadri- lateral is formed by hinging two triangles along a common edge. So, for any quadrilateral, we can change the angle between the two triangles to either 0 or tt to flatten the quadrilateral, that is planar. So, Geo{4) is path-connected. • Another way to prove Theorem 5.3.2 is to use the following lemma.

Lemma 5.3.3. Let P =< vi,v2, • • • , �be an n-sided polygon embedded in and let Aj be the union of the interior of the triangular disc with vertices {t;i_i，Vi’tii+i} and the line segment Vi-iVi+i, where i is the mod n index. If P 门 Ai = 0 for some i, then there is a path in Geo(n) that connects P to an (n - l)-sided polygon P' =< Vi,V2, • • • ’ .. • ,Vn > (since (n - l)-sided polygon can be viewed as a special case of n-sided polygon). Proof. Let h : [0,1] — K^n be the path which corresponds to the movement that move the vertex Vi in a straight line path across Ai to the midpoint of ViVi+i and fixes the other n - 1 vertices of P. That is h(f) =< Vi,V2,--' ,Vi-uVi(l-t) + I…i+l + ?;i-i)’"?^i+i’... ,Vn> . Geometric Knot Theory 42

Now, since _P n Ai = 0，this isotopy does not introduce any intersection. So, h is actually the path in Geo{n) which is needed. • We now use this lemma to prove (Theorem 5.3.2). Proof. (Theorem 5.3.2) Let P =< v^, t;4 > be a polygonal knot in Geo(4), let A2 be as defined in Lemma 5.3.3. So we have i^A A2 = 0’ by Lemma 5.3.3, there is a path in Geo(4) connecting P to the triangle P' =< ？;1,> which is planar. Now, since P is arbitrary, Geo(4) is really path connected. • Finally, we consider the space Geo(5).

Theorem 5.3.4. Geo(5) is path-connected.

Proof. Let P =< vuv2, vz, v^, > be a polygonal knot in Geo(5). If Pn A2 = 0, then, by Lemma 5.3.3, P is geometric equivalent to P' =< i^i,i>3，"4，幻5� By. Theorem 5.3.2, P' is a geometric unknot, so P is also a geometric unknot. If P n A2 0, this means the edge v^v^ passes through A2. Then we must have P 门 A5 = 0 (see Figure 5.1), hence, by Lemma 5.3.3 and Theorem 5.3.2 again, P is also a geometric unknot. So Geo(5) is path-connected. •

•之A

Figure 5.1: The case of P n A2 0. Geometric Knot Theory 43 5.4 Topology of the Space Geo(6) In this section, we will focus our discussion on the space Geo(6). In chapter 3，we have known that the trefoil knots can be visualized with six sticks. Also, it is well known that the left trefoil and right trefoil are not isotopic. So, combining this facts, we know that Geo(6) must contain at least three path-components which correspond to the unknot, the left trefoil and the right trefoil. In fact, We have the following theorem.

Theorem 5.4.1. The space Geo(6) has five path-components. These consist of one component of the topological unknot, two components of the left trefoil knots and two compo- nents of the right trefoil knots.

In order to prove this theorem, we need some preparations. Firstly, by transla- tions and rotations, any hexagonal knot can be replaced by a geometric equivalent knot of the form H =< Vi,V2, v^, V4, V5, vq > with Vi = (0,0’ 0) and V5 = (x^, 0,0), X5 > 0. Next, we assume that H intercepts the x-axis only at vi and v^. Let P2, Ps and P4 be the half-plane starting from the a;-axis and containing V2, v^ and v^, respectively. We add one more assumption that P2, and P4 are distinct. Now, we choose a half-plane starting from x-axis which does not intersect H. Then we rotate it around the a;-axis in the right-handed fashion. This define an ordering for the half-planes P2, P3 and P4. There are six cases: 2-3-4, 2-4-3, 3-2-4, 3-4-2, 4-2-3 and 4-3-2, where 2-3-4 corresponds to the ordering P2 P3 Pa and etc. Then, we will analyze these six cases and get the following lemma.

Lemma 5.4.2. Each of these cases gives a single region of unknot. Each of the cases 2-4-3 and 3-2-4 also gives a single region of right trefoil. And each of the cases 3-4-2 and Geometric Knot Theory 44

4-2-3 also gives a single region of left trefoil.

Proof. We can assume that ？;2, v^ and v^ are linearly independent. If not, by moving a little, we can replace H by a. geometric equivalence knot. Similarly, we can also assume that Vq dose not lie in the half-planes 尸2，Ps or P4. Next, by rotation, we can assume V2 lies in the upper-half xy-p\a,ne. Combining these with the fact that V4 does not lie in the a:y-plane, we can always find a linear transformation A with det A > 0, which fixes V4 and V5 and maps V2 to (臺狗,1,0). Since det A > 0, there exists a path p : [0,1] — GL(3,M) such that p(0)=I and p(l)=A. So, by defining At = p(f), we get a continuous family of linear transformations which deform H to a. geometric equivalence knot with ZV5V1V2 acute. Then, by a similar transformation (fixing V2 and ？;5), we can deform the resulting knot to a geometric equivalence knot with ZV1V5V4 acute. So, we can fix a projection of H such that V2 lies on the right of ^；! and V4, lies on the left of f5. Then, we can analyze the six cases.

(1) 2-3-4 In this case, every half-plane between P2 and P4 intersects the union of two edges, V2V3 U V3V4, at exactly one point. We have the following cases:

• If Vq does not lie in these half-plane, then 丑 n A5 = 0. • If Vq lies in a half plane between P2 and P3, then 丑 n A4 = 0. • If Vq lies in a half plane between P3 and P4, then /J n A2 = 0.

Hence, by Lemma 5.3.3，H is always geometric equivalent to a pentag- onal knot. By Theorem 5.3.4, we know that each pentagonal knot is a geometric unknot, so, H is always a geometric unknot. So, 2-3-4 only gives a single region of unknot. (2) 2-4-3 Geometric Knot Theory 45

In this case, each half-plane between P2 and P4 intersects the edge V2V3 at a single point, and each half-plane between P2 and P4 intersects each of the edges V2VS and V3V4 at exactly one point. Next, we consider the position of the two-edges linkage v^vevi. If V5VQV1 dose not pass through the triangle A3 which is formed by the vertices V2, v^ and V4, then we have 丑 A A3 = 0 and Lemma 5.3.3 is applied, this implies H is & geometric unknot. If V5VQV1 passes through A3. This means vq must lie in a half-plane between P2 and P3. If Vq lies in a half-plane between P2 and P4, then we have H 0 A4 = (/} and again iJ is a geometric unknot. So, we only remain the case of v^vqVi passing through A3 and lying in a half-plane between P4 and P3. We consider the three possible configurations of the four-edges linkage V1V2VSV4V5, which are shown in Figure 5.2. Note that, we can always choose a projection such that V2 and V4 are separated by the x-axis. So, by definition of 2-4-3’ the edge V2V3 must cross over the rc-axis. Z� Vi j V5 Vi V5 Vi ^ \ V5 V3 W V3> V3>

⑷ (b) (c)

Figure 5.2: The configurations of viV2VzVav^ in the case 2-4-3.

(a) V2V^ intersects P4 at the left of (see Figure 5.2(a)). We have the following cases: • If V5VQ passes through A3, then we have 丑 n A4 = 0. • If vqVi passes through A3, then we have H D A<2 = (h Geometric Knot Theory 46

• If ve lies on A3, then we have both 丑 n A2 = 0 and 丑 n A4 二 0. Hence, Lemma 5.3.3 can always be applied and we get H is sl geo- metric unknot. (b) V2V3 intersects P4 at the right of V4 such that V2V3 and V4V5 form a negative crossing^ that is V2V3 passes under V4V5 (see Figure 5.2(b)). Then, we must have one of the following two cases: (i) The linkage VsVeVi passes over the edge V2V3 and then under the linkage (ii) The linkage V5V6V1 passes under the edge V2Vs and then over the linkage V3V4V5. For (i), we have the following cases: • If V5V6 passes through A3, then we have H Ct A4 = 0. • If vevi passes through A3, then we have H H A2 = 0. • If V6 lies on A3, then we have both if n A2 = 0 and 丑门 A4 = 0. Hence, Lemma 5.3.3 can always be applied and we get H is a geo- metric unknot. For (ii), we consider the plane P determined by V2, V3 and V4. Again, by moving a little if needed, P can always be uniquely defined. Now, since x-axis lies below A3 and the linkage v^v^vi needs to pass through A3 with undercrossing the edge V2Vz which lies under the vertex v^, the linkage vf^v^vi must cross P at least three times, which is impossible. So, H must be a geometric unknot in this configuration. (c) V2VS intersects P4 at the right of V4 such that V2V3 and v^v^ form a positive crossing^, that is V2Vs passes over v^v^ (see Figure 5.2(c)). 1 negative crossing means the overcrossing arc go from the right to the left of the undercross- ing arc with the given orientation. 2 positive crossing means the overcrossing arc goes from the left to the right of the under- Geometric Knot Theory 47

Then, we must have one of the following two cases: (i) The linkage v^vqVi passes over the edge V2V3 and then under the linkage V3V4V5. (ii) The linkage VsVqVi passes under the edge V2V3 and then over the linkage V3V4V5. (ii) is basically the same as (b)(i). Hence, we get H is & geometric unknot. For (i), it is clearly that if such hexagonal knots exist then all of them geometric equivalent. So, we only need to show the existence. Actually, we can choose H as following, 90 C pp H =< (0，0’ 0), (2’ 1’ 0), (3，-2,1), (2, -1’-), (4’ 0’ 0), (-, -1’ -) > . Then, by projecting to the rcy-plane, we get Figure 5.3.

Vi \ Li^ V5

Figure 5.3: An example of right trefoil in 2-4-3.

So, in the case 2-4-3’ we have one region of unknot (the geometric unknot) and one region of right trefoil. (3) 3-2-4 In order to analysis this case, we first rotate H about the line x = -X5 with angle tt and then change the orientation by relabeling the vertices with the crossing arc with the given orientation. Geometric Knot Theory 48

following, v'l = r(v5), v'2 = r(v4), = 7>3)’ V4 = r(v2), v'5 = r(vi), v'q = r(v6), where r is the given rotation. Now, H' =< vi,v'2,v'3,v'4,v'5’v'6 > is actually the type of knot in case 2-4-3. So, H' is an unknot or a right trefoil. Since our process does not change the isotopic type of knot, H must also be an unknot or a right trefoil. So, in the case 3-2-4, we have one region of unknot (the geometric unknot) and one region of right trefoil. (4) 3-4-2 In this case, after a reflection through the plane determined by Vi, and V5, we can change a 3-4-2 knot to a 2-4-3 knot. Also, the reflection of an unknot is an unknot and the reflection of a right trefoil is a left trefoil, so, in the case 3-4-2, we have one region of unknot (the geometric unknot) and one region of left trefoil. (5) 4-2-3 In this case, after a reflection through the plane determined by vi, V2 and ？;5, we can change a 4-2-3 knot to a 3-2-4 knot. Also, the reflection of an unknot is an unknot and the reflection of a right trefoil is a left trefoil, so, in the case 4-2-3’ we have one region of unknot (the geometric unknot) and one region of left trefoil. (6) 4-3-2 In this case, after a reflection through the plane determined by Vi, V3 and V5, we can change a 4-3-2 knot to a 2-3-4 knot. Also, the reflection of an unknot is an unknot, so, in the case 4-3-2, we have only region of unknot (the geometric unknot). Geometric Knot Theory 49

Hence, we finish our analysis on the six cases and the results are summarized by the Table 5.1. •

Region Unknot Right trefoil Left trefoil 2-3-4 1 0 0 2-4-3 1 1 0 3-2-4 1 1 0 3-4-2 1 0 1 4-2-3 1 0 1 4-3-2 1 0 0 Table 5.1: Number of path-components in each region

Now, we can prove Theorem 5.4.1. Proof. (Theorem 5.4.1) Actually, it remain to prove the following facts: (1) The in the six cases are all geometric equivalent, that is they are all geometric unknot. (2) The right trefoil knots in 2-4-3 are not geometric equivalent to the right trefoil knots in 3-2-4. (3) The left trefoil knots in 3-4-2 are not geometric equivalent to the left trefoil knots in 4-2-3. In Lemma 5.4.2, we have proved all the unknots in each case that can be deformed to the pentagons which are all proved to be the geometric unknot by Theorem 5.3.4. So, we finish the proof (1). In order to prove (2) and (3), we consider the f : Geo{Q)—> M, which is defined by f{H) = {{V5 - Vi) X {V2 - Vl)) . - ^l), Geometric Knot Theory 50 where H =< 1*2’ 仍，幻4,幻5,询 Geo(6). For the function /, we have the following lemma.

Lemma 5.4.3. If if is a knot such that f(H) 二 0，then H is an unknot (actually, a geometric unknot). Proof. If 丑 is a knot such that f{H) = 0, then Vi, V2, V3 and v^ all lie in the xy-plane. We have the following cases: (a) If both of V4 and ve also lie in the rcy-plane, then H is planar which is an unknot. (b) If ？;4 lies in the rcy-plane and vq does not, then there are two subcases: (i) If v^ or v^ lie inside A2, then, by applying Lemma 5.3.3 to A4 and applying Theorem 5.3.4，H is an unknot. (ii) If V4, and v^ do not lie inside A2, then, by applying Lemma 5.3.3 to A2 and applying Theorem 5.3.4, H is an unknot. (c) If Vq lies in the rc?/-plane and V4 does not, similar to (b), again H is an unknot. (d) If both of V4 and vq do not lie in the x^-plane, then there are two subcases: (i) If V5 lies inside A2, then, by applying Lemma 5.3.3 to A5 and ap- plying Theorem 5.3.4, H is an unknot. (ii) If V5 do not lie inside A2, then, by applying Lemma 5.3.3 to A2 and applying Theorem 5.3.4, If is an unknot. Hence, in all cases, H must be an unknot • Geometric Knot Theory 51

Now, we recall that V2 can be assumed to be lying in the upper xy-plane, so if i/ is a knot of case 2-4-3 or 4-2-3, then V3 must lie above the xy-plane, that is f{H) > 0. However, if if is a knot of case 3-2-4 or 3-4-2, then vs must lie below the a^y-plane, that is f{H) < 0. Next, we prove (2) by contradiction. Suppose the right trefoil knots in 2-4-3 are geometric equivalent to the right trefoil knots in 3-2-4. So, there exists a path h : [0’ 1] — Geo{Q) such that h(0) is a right trefoil knot of case 2-4-3 and h{l) is a right trefoil knot of case 3-2-4. Then, we consider the continuous function foh: [0,1] ^R, we have foh{0) > 0 and/o/i(l) < 0. So, there exists t e (0，1) such that foh{t) = 0’ this implies that the right trefoil Ht = h[t) has f{Ht) = 0， which contradicts to Lemma 5.4.3. So, we finish the proof of (2). And the proof of (3) is just similar to the proof of (2), so the whole proof of Theorem 5.4.1 is finished. • Actually, Theorem 5.4.1 gives examples (the left trefoil and the right trefoil) that isotopic equivalence does not imply the geometric equivalence. This means that the geometric equivalence is stronger then the isotopic equivalence Chapter 6 Concluding Remarks

In this thesis, we have discussed three new developments in the knot theory. Actually, in each topic, there are some results that we have not covered. In this chapter, we will introduce those results (without proof) and we will discuss some possible directions of further developments.

6.1 Other Results on the Minimal Stick Num- ber We first discuss the minimal stick number. In chapter 3’ we use the crossing number to give the bounds to the minimal stick number. Actually, if we only consider a specific type of knots, the 2-bridge knots^ then we can improve the upper bound. In McCabe's paper [16], she has proved the following,

Theorem 6.1.1. For any 2-bridge knot K, the inequality slK]

Clearly, only considering the 2-bridge knot, this upper bound is better than the upper bound of the equation (3.9) in chapter 3. Actually, in her paper, she directly constructs the polygonal representation of K with c[K] + 3 edges. And her construction is based on the fact that the class of 2-bridge knot is exactly the class of rational knot (using tangle) which was first introduced by Conway in [8]. A similar result about the rational knot can be found in [10]. In order to state it, we need to introduce the Conway notation. In fact, every rational knot can be expressed as a continued fraction which corresponds to the tangles making the knot, this continued fraction is called the Conway notation representation of the corresponding knot. Actually, it can be proved that two rational knots are isotopic equivalent if and only if their continued fractions represent the same rational number (proof can be found in [4]). This also gives a bijection between the classes of rational knot and the rational numbers between 0 and 1. Now, we can conclude the result in [10] as following,

Theorem 6.1.2. For any rational knot K which has a 1-, 2- or 3-Conway integer notation repre- sentation and c(K) > 6’ we have the inequality s[K]

The proof of this theorem is an induction on the crossing number and is based on the fact that the corresponding tangles of n-Conway integer knot always display the crossing number. We present the steps here. We consider the 1- Conway integer knot. The first 1-Conway integer knot satisfying our conditions is a knot with crossing number 7 which can be constructed with nine edges. Then, we find out the free vertex in the nine edges figure and use it to form a 1-Conway Geometric Knot Theory 54 integer knot with crossing number 9 with 2 more edges. By repeating the process inductively, we finish the proof of 1-Conway integer knot. And the proof for 2- or 3-Conway integer knot is similar, but more initial knots are needed. Since all rational knots can be expressed as the n-Conway integer notation for some n, the idea of the proof may apply for any rational knot. This is certainly not an ideal way to prove that the upper bound works for all rational knots and, in fact, the general case is still an open question.

6.2 Minimal Stick Number and Superbridge In- dex of the Torus Knot For any pair of integers p and q with p,q>2 and gcd(p,q) = 1, the (p,g)-torus knots are the knots being equivalent to the knot given by the parametrization, x{t) = cospt(2 + cos qt) < y ⑷二 sinp啦+ cosgi) �z{t) 二 singt, which lies on the unknotted torus with parametrization, X = cos ^(2 +cos 0) < y = sin ^(2 +cos 0) �z = smcj). Now, it is known that for any pair of p and q, the (p, g)-torus knot and the (g,p)-torus knot are equivalent, so we can only consider the case ofp < q. For the superbridge index of the torus knots, Kuiper [12] proved the following,

Theorem 6.2.1. For any pair of relatively prime integers p, q with 2 < p < q, the superbridge Geometric Knot Theory 55 index of the (p, g)-torus knot K is sb[Kp^g] = mm{q,2p).

Next, we consider the minimal stick number of the torus knots. Jin [11] proved the following,

Theorem 6.2.2. For any pair of relatively prime integers p, q with 2 < p < g, the minimal stick number of the (p, ^)-torus knot K is

< 2q.

Proof. Let aG min(7r’ and 々二宇一a, so we have

max(0, ^ -TT�

Theorem 6.2.3. For any pair of relatively prime integers p, q with 2 < p < g, the minimal stick number of the (p, g)-torus knot K is min(25,4p) < < 2q. Geometric Knot Theory 56

In particular, if 2 < p < ^ < then we have

Hence, the superbridge index and minimal stick number are known for an infinite class of knots.

6.3 Explorations of the Geometric Knot Spaces In chapter 5’ we have studied the geometric knot spaces, Geo(n), for n = 3,4,5’ 6. In [5], Calvo also completely studied the case of Geo(7) and had a partial result on Geo(8). His results can be summarized as following,

Theorem 6.3.1. The space Geo(7) has five path-components. These consist of one component of the topological unknot, one component of the left trefoil knot, one component of the right trefoil knot and two components of the figure-eight knot.

Theorem 6.3.2. The complete list of topological knots in Geo(8) contains all knots with crossing number smaller than or equal to six and two of the crossing number 8 knots, 8i9 and 820- Since every knot, with the exception of the unknot, the figure-eight knot, 63 and the connected sum of left trefoil and right trefoil, are not isotopic equivalent to their mirror image, Geo(8) has at least twenty path-components.

Actually, as Theorem 6.3.2, the geometric knot spaces Geo(n), n > 8, are extremely complex and difficult to study directly. In order to explore these spaces, one method is the Metropolis Monte Carlo sampling. In the following, we will discuss how it works. Firstly, we observe that multiplying every coordinate with a constant preserves Geometric Knot Theory 57 the geometric knot types. So, every geometric knot is geometric equivalent to one whose vertices lie in the unit closed ball of with at least one vertex lies in the boundary of the closed ball. Now, we randomly select n-points on the unit sphere with respect to the uniform distribution, n - 1 radii with respect to the uniform distribution on [0’ 1] and a number i with respect to the uniform distribution on the set (1, • • • ,n}. Then we allow the ith vertex to have magnitude one and the others lie inside the unit closed ball with magnitude determined by the radii above. This forms a polygon which represents a class of polygon in with the ith vertex having the largest magnitude. Secondly, we estimate its knot type of the random knot by calculating the HOMFLY polynomial which can distinguish most of the knots up to their isotopic types. Finally, since any geometric knot is geometric equivalent to one whose vertices correspond to a point in the unit sphere of the proportion of a geometric knot type of Geo(n) is equal to the proportion of that lying on the unit sphere of SO, by considering the corresponding point of the random knot in the unit sphere of R^", we can estimate the proportion of the corresponding HOMFLY polynomial in Geo{n). This sampling gives us a brief description of the structure of the space Geo(n) such as the approximated number of distinct isotopic knot types and the propor- tion of the isotopic knot types. Also, it helps us to determine the minimal stick number of some knots. For exmaple, if a certain knot type is found in the space G?eo(9) and it is not in the complete list of Theorem 6.3.2’ then the minimal stick number of that knot type must be 9. Bibliography

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