THESIS IMPROVED STICK NUMBER UPPER BOUNDS Submitted by Thomas D. Eddy Department of Mathematics In partial fulfillment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Spring 2019 Master’s Committee: Advisor: Clayton Shonkwiler Henry Adams Hamid Chitsaz Copyright by Thomas D. Eddy 2019 All Rights Reserved ABSTRACT IMPROVED STICK NUMBER UPPER BOUNDS A stick knot is a mathematical knot formed by a chain of straight line segments. For a knot K, define the stick number of K, denoted stick(K), to be the minimum number of straight edges necessary to form a stick knot which is equivalent to K. Stick number is a knot invariant whose precise value is unknown for the large majority of knots, although theoretical and observed bounds exist. There is a natural correspondence between stick knots and polygons in R3. Previous research has attempted to improve observed stick number upper bounds by computationally generating such polygons and identifying the knots that they form. This thesis presents a new variation on this method which generates equilateral polygons in tight confinement, thereby increasing the incidence of polygons forming complex knots. Our generation strategy is to sample from the space of confined polygons by leveraging the toric symplectic structure of this space. An efficient sampling algorithm based on this structure is described. This method was used to discover the precise stick number of knots 935, 939, 943, 945, and 948. In addition, the best-known stick number upper bounds were improved for 60 other knots with crossing number ten and below. ii ACKNOWLEDGEMENTS The foremost person I must thank for helping me complete this thesis is my advisor, Clay Shonkwiler. He is responsible for suggesting this interesting and fruitful topic for my master’s thesis. Our many hours of discussion have always been insightful and enjoyable. Clay’s guidance has been invaluable thus far in my graduate education. I am indebted to Harrison Chapman for generously aiding me on several occasions. With- out his help debugging code, there are many points where this research could have stalled. Thanks also for critiquing an early version of a talk, being a source of random polygon knowledge, and giving advice about academic life. I’m also grateful to Jason Cantarella for giving me access to the plCurve project repos- itory. Allowing me to branch the project made the results of this thesis possible. I would be remiss if I didn’t mention Colin Roberts and Brenden Balch who are always willing to talk about geometry with me. Plenty of the things I understand about symplectic geometry have come from grappling with these ideas at Colin’s whiteboard. Additionally, they provided thoughtful feedback on the first draft of this thesis. Finally, I am appreciative for the tremendous amount of love and support that my parents continue to share with me. A phone call with them is often the perfect antidote to a frustrating day. iii DEDICATION This thesis is dedicated to Juanita Duque Rosero. iv TABLE OF CONTENTS ABSTRACT . ii ACKNOWLEDGEMENTS . iii DEDICATION . iv LIST OF FIGURES . vi Chapter 1 Introduction . 1 Chapter 2 Background . 3 2.1 Knot Theory . 3 2.2 Symplectic Geometry . 20 Chapter 3 Generating Random Polygons in Confinement . 25 3.1 The Space of Confined Equilateral Polygons . 25 3.2 Algorithm for Sampling Confined Polygon Space . 31 Chapter 4 Results . 36 4.1 Methodology . 37 4.2 Improved Stick Number Bounds . 40 4.3 Further Questions . 44 Bibliography . 47 Appendix A Stick Number Bounds . 51 A.1 Upper Bounds for Small Crossing-Number Knots . 52 A.2 Upper Bounds for Large Crossing-Number Knots . 58 Appendix B Knot Frequency Counts . 68 B.1 Frequency of Knots Generated in Confinement . 69 Appendix C Coordinates of Discovered Stick Knots . 96 C.1 9-Stick Knots . 97 C.2 10-Stick Knots . 98 C.3 11-Stick Knots . 115 Appendix D Index . 173 v LIST OF FIGURES 2.1 Trefoil knot with knot diagram . 4 2.2 Trefoil knot deformation . 5 2.3 Reidemeister moves . 6 2.4 Stick knot . 7 2.5 Knot composition . 8 2.6 Torus knot . 9 2.7 Torus knot diagrams . 9 2.8 Simple knots . 11 2.9 Trefoil knot decomposition . 13 2.10 Vector field on R2 generated by translation . 22 2.11 Vector field on S2 generated by a Hamiltonian action . 23 3.1 Fan triangulation coordinates . 27 3.2 Pol3(6) polytope . 28 3.3 Polygon in rooted confinement . 29 d 3.4 Pol3,1(6) polytope . 30 3.5 TSMCMC polytope update step . 33 d 4.1 Equilateral 9-stick knots . 41 4.2 Equilateral 10-stick 10147 knot . 42 4.3 Equilateral 10- and 11-stick knots . 43 vi Chapter 1 Introduction The stick number of a knot K is the least number of straight sticks which can be chained together to form K. This number is a knot invariant which has been studied since at least the 1990’s [33]. Although the invariant can be easily and intuitively defined, relatively little is known about the precise stick number for most knot types. Prior to this thesis, stick number was known for only 31 of the 250 knots with crossing number ten or fewer, as can be observed in Section A.1. Nonetheless, various theoretical and observed bounds for stick number exist. Many of the stick numbers which are known precisely have come from lowering the observed upper bounds until they became equivalent to the theoretical lower bounds. The most successful reductions in observed upper bounds have typically come from computational means, as in [34]. This thesis presents a new computational method for further reducing observed stick number upper bounds. Our strategy starts by considering stick knots as polygons in three- dimensional space. We try to randomly generate many such n-gons hoping that we will observe a sample forming a knot whose best-known stick number upper bound is greater than n. If so, this sample represents an improvement to the upper bound. We do not naively sample polygons, however. We consider an enriched sample of polygons in spherical confinement. Intuitively, sampling from the space of confined polygons should increase the chance that the samples form more complex knots. As we can observe in Appendix B, the formation of complex knots is still rare in con- finement; trivial knots predominate. Even so, sampling in this way does increase our odds of observing complex knots. Indeed, this method proved very effective at improving stick number upper bounds. After sampling 40 billion stick knots from the space of confined 1 polygons, we were able to improve the best-known bounds of 65 knots with crossing number ten and below. Moreover, these reductions in upper bounds contributed to the discovery of the precise stick number of five knots. A full summary of results is presented in Chapter 4. Chapter 2 provides definitions, background, and previous results relevant to the text. The chapter starts with a self-contained introduction to knot theory, focusing on the stick number invariant and methods for identifying knots. A review of selected topics from sym- plectic geometry follows. This material is necessary to describe the algorithm used to sample confined polygon space. Chapter 3 details the algorithm, originally proposed in [13], we used to generate confined random polygons. This algorithm is based on the toric symplectic structure of confined polygon space. The chapter begins by describing this structure and the special properties it confers. The remainder of the chapter is devoted to a description of the algorithm, including pseudo-code. The goal is for the reader to be able to understand and implement the algorithm based on this description. The final chapter presents the new stick number results mentioned above. In the first section we discuss the exact methodology used to generate polygons and identify their knot type. The main results of the thesis follow. The text concludes with a discussion of remaining questions and open problems. Additional avenues of research are suggested. Detailed appendices are also included with the text. Appendix A contains tables listing the best-known upper bounds on stick number. All knots with crossing number ten and below are included, as well as bounds for selected knots with more crossings. Appendix B lists the frequency counts of sampled n-gons. Appendix C contains vertex coordinates for each knot type for which an improved bound was discovered. Finally, an index of important terms is available in Appendix D. 2 Chapter 2 Background In order to understand the polygon generation algorithm described in Chapter 3 and give context to the results presented in Chapter 4, we first review some prerequisite concepts. This material falls roughly into two disciplines: knot theory and symplectic geometry. Section 2.1 gives a self-contained introduction to the topics in knot theory which are referenced later in this thesis. Specifically, the section will cover: basic concepts and def- initions; important classes of knots; knot invariants; and previous results related to stick number. Stick knots are the essential objects of study in this thesis and thus it is important for the reader to have a clear understanding of this material. In contrast, the overview of relevant symplectic geometry concepts provided in Section 2.2 is briefer. This is due to the fact that these ideas are less central to the results of this thesis and because they require substantial background knowledge. Indeed, an understanding of the fundamentals of differential geometry is assumed. This section is included because it is necessary to understand the algorithm for randomly sampling confined polygons presented in Chapter 3. If the reader is only interested in the Chapter 4 results, then this section can safely be skipped.