Geometric Knot 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 Knot Theory i Abstract The space of n-sided (n > 3) polygonal knots embedded in consists of a smooth manifold 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 of five 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 Topology 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 Torus Knot 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 embedding 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, homeomorphisms 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 unknot 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 homeomorphism 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.