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Knots, Tangles and Braid Actions KNOTS, TANGLES AND BRAID ACTIONS by LIAM THOMAS WATSON B.Sc. The University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : THE UNIVERSITY OF BRITISH COLUMBIA October 2004 c Liam Thomas Watson, 2004 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for ex- tensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Mathematics The University of British Columbia Vancouver, Canada Date Abstract Recent work of Eliahou, Kauffmann and Thistlethwaite suggests the use of braid actions to alter a link diagram without changing the Jones polynomial. This technique produces non-trivial links (of two or more components) having the same Jones polynomial as the unlink. In this paper, examples of distinct knots that can not be distinguished by the Jones polynomial are constructed by way of braid actions. Moreover, it is shown in general that pairs of knots obtained in this way are not Conway mutants, hence this technique provides new perspective on the Jones polynomial, with a view to an important (and unanswered) question: Does the Jones polynomial detect the unknot? ii Table of Contents Abstract ii Table of Contents iii List of Figures v Acknowledgement vii Chapter 1. Introduction 1 Chapter 2. Knots, Links and Braids 3 2.1 Knots and Links : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Braids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 Chapter 3. Polynomials 8 3.1 The Jones Polynomial : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3.2 The Alexander Polynomial : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.3 The HOMFLY Polynomial : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 Chapter 4. Tangles and Linear Maps 21 4.1 Conway Tangles : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 4.2 Conway Mutation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4.3 The Skein Module : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 4.4 Linear Maps : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 4.5 Braid Actions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32 Chapter 5. Kanenobu Knots 37 5.1 Construction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37 5.2 Basic Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 5.3 Main Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 5.4 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 46 5.5 Generalisation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50 5.6 More Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 53 5.7 Observations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 iii Table of Contents iv Chapter 6. Thistlethwaite Links 62 6.1 Construction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 62 6.2 Some 2-component examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69 6.3 A 3-component example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 70 6.4 Closing Remarks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 Bibliography 72 iv List of Figures 2.1 Diagrams of the Trefoil Knot : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 The Hopf Link : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 The braid generator σi and its inverse : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.4 The link β¯ formed from the closure of β. : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 4.1 Some diagrams of Conway tangles : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 4.2 The generator ei : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 4.3 The tangle T β : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 33 2 S2 5.1 The Kanenobu knot K(T; U) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37 5.2 Distinct knots that are not Conway mutants : : : : : : : : : : : : : : : : : : : : : : 43 5.3 Example 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 47 5.4 Example 2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 49 5.5 Another diagram of the Kanenobu knot K(T; U) : : : : : : : : : : : : : : : : : : 50 5.6 The link β : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50 j j 5.7 The closure of a Kanenobu braid : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51 3 −1 5.8 The braid σ1σ2 σ1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 53 3 −1 −3 −1 5.9 The Kanenobu braid (σ1σ2 σ1)(σ5 σ4σ5 ) : : : : : : : : : : : : : : : : : : : : : : : 53 3 −1 −3 −1 5.10 The knot (σ1σ2 σ1)(σ5 σ4σ5 ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 54 ? 5.11 The knot K(0; 0) 52 # 52 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 55 −∼1 5.12 The knot K (σ2; σ2 ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 55 2 −3 5.13 The braid σ1σ2 σ1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56 2 −3 −2 3 −1 5.14 The Kanenobu braid (σ1σ2 σ1)(σ5 σ4σ5 ) : : : : : : : : : : : : : : : : : : : : : : : 56 2 −3 −2 3 −1 5.15 The knot (σ1σ2 σ1)(σ5 σ4σ5 ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 57 ? 5.16 The knot K(0; 0) 61 # 61 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 −∼1 5.17 The knot K (σ2; σ2 ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 5.18 The 2-bridge link obtained from β B3. : : : : : : : : : : : : : : : : : : : : : : : : : : 59 ? 2 5.19 The square knot 31 # 31 . : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60 −1 −1 −1 5.20 The knot (σ1σ2 σ1)(σ5 σ4σ5 ) . : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60 6.1 The Thisleth waite link H(T; U) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 6.2 The link H(0; 0) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 6.3 The braid ! B3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 66 2 −1 6.4 The link H(T !; U ! ) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 67 6.5 The result of !2 acting on H(T; U) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68 6.6 The result of !4 acting on H(T; U) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68 v List of Figures vi 6.7 A non-trivial, 2-component link : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69 6.8 A non-trivial, 2-component link : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 70 6.9 A non-trivial, 3-component link : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 vi Acknowledgement It's a strange process, writing down the details to something you've been work- ing on for some time. And while it is a particularly solitary experience, the final product is not accomplished on you're own. To this end, there are many people that should be recognised as part of this thesis. First, I would like to thank my supervisor Dale Rolfsen for ongoing patience, guidance and instruction. It has been a privilege and a pleasure to learn from Dale throughout my undergraduate and masters degrees, and I am grateful for his generous support, both intellectual and financial. I have learned so much. Of course, I am indebted to all of my teachers as this work comprises much of what I have learned so far. However, I would like to single out Bill Casselman, as the figures required for this thesis could not have been produced
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