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El. knyga: Diagram Genus, Generators, and Applications [Taylor & Francis e-book]

(GIST College, Gwangju Institute of Science and Technology, South Korea)
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Diagram Genus, Generators, and ApplicationsDidesnis paveikslėlis
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In knot theory, diagrams of a given canonical genus can be described by means of a finite number of patterns ("generators"). Diagram Genus, Generators and Applications presents a self-contained account of the canonical genus: the genus of knot diagrams. The author explores recent research on the combinatorial theory of knots and supplies proofs for a number of theorems.

The book begins with an introduction to the origin of knot tables and the background details, including diagrams, surfaces, and invariants. It then derives a new description of generators using Hirasawas algorithm and extends this description to push the compilation of knot generators one genus further to complete their classification for genus 4. Subsequent chapters cover applications of the genus 4 classification, including the braid index, polynomial invariants, hyperbolic volume, and Vassiliev invariants. The final chapter presents further research related to generators, which helps readers see applications of generators in a broader context.
Preface ix
List of Figures
xiii
List of Tables
xv
Symbol Description xvii
1 Introduction
1(12)
1.1 The beginning of knot theory
1(2)
1.2 Reidemeister moves and invariants
3(1)
1.3 Combinatorial knot theory
4(1)
1.4 Genera of knots
5(1)
1.5 Overview of results
6(4)
1.5.1 General generator estimates
6(1)
1.5.2 Compilation of knot generators of genus 4
6(1)
1.5.3 Unknot diagrams
7(1)
1.5.4 Non-triviality of the link polynomials
8(1)
1.5.5 Signature of positive knots
8(1)
1.5.6 Braid index and Bennequin surfaces
9(1)
1.5.7 Properties of the Alexander polynomial of alternating knots
10(1)
1.6 Issues of presentation
10(1)
1.7 Further applications
11(2)
2 Preliminaries
13(34)
2.1 Knots and diagrams
13(2)
2.2 Crossing number and writhe
15(1)
2.3 Knotation and not-tables
16(2)
2.4 Seifert surfaces and genera
18(4)
2.4.1 Special diagrams
21(1)
2.5 Graphs
22(3)
2.5.1 General definitions
22(2)
2.5.2 The checkerboard coloring and checkerboard graph
24(1)
2.6 Diagrammatic moves
25(3)
2.6.1 Flypes and mutations
25(2)
2.6.2 Bridges and wave moves
27(1)
2.7 Braids and braid representations
28(1)
2.8 Link polynomials
29(7)
2.8.1 Generalities
29(1)
2.8.2 Skein, Jones, and Alexander polynomial
30(3)
2.8.3 Kauffman polynomial and semiadequacy
33(3)
2.9 MWF inequality, Seifert graph and graph index
36(2)
2.10 The signature
38(3)
2.11 Genus generators
41(4)
2.12 Knots vs. links
45(2)
3 The maximal number of generator crossings and ~ -equivalence classes
47(16)
3.1 Generator crossing number inequalities
47(1)
3.2 An algorithm for special diagrams
48(6)
3.3 Proof of the inequalities
54(3)
3.4 Applications and improvements
57(6)
4 Generators of genus 4
63(6)
5 Unknot diagrams, non-trivial polynomials and achiral knots
69(30)
5.1 Some preparations and special cases
69(2)
5.2 Reduction of unknot diagrams
71(4)
5.3 Simplifications
75(7)
5.4 Examples
82(2)
5.5 Non-triviality of skein and Jones polynomial
84(4)
5.6 On the number of unknotting Reidemeister moves
88(5)
5.7 Achiral knot classification
93(6)
6 The signature
99(4)
7 Braid index of alternating knots
103(16)
7.1 Motivation and history
103(1)
7.2 Hidden Seifert circle problem
104(1)
7.3 Modifying the index
105(5)
7.4 Simplified regularization
110(5)
7.5 A conjecture
115(4)
8 Minimal string Bennequin surfaces
119(8)
8.1 Statement of result
119(1)
8.2 The restricted index
120(1)
8.3 Finding a minimal string Bennequin surface
121(6)
9 The Alexander polynomial of alternating knots
127(18)
9.1 Hoste's conjecture
127(6)
9.2 The log-concavity conjecture
133(5)
9.3 Complete linear relations by degree
138(7)
9.3.1 Linear relations up to constants
138(3)
9.3.2 General genus
141(1)
9.3.3 Removing inaccuracy up to constants
141(4)
10 Outlook
145(14)
10.1 Legendrian invariants and braid index
145(2)
10.2 Minimal genus and fibering of canonical surfaces
147(2)
10.3 Wicks forms, markings, enumeration of alternating knots by genus
149(3)
10.4 Crossing numbers
152(1)
10.5 Canonical genus bounds hyperbolic volume
153(3)
10.6 The relation between volume and the sln polynomial
156(2)
10.7 Everywhere equivalent links
158(1)
Bibliography 159(8)
Glossary 167(4)
Index 171
Alexander Stoimenow is an assistant professor in the GIST College at the Gwangju Institute of Science and Technology. He was previously an assistant professor in the Department of Mathematics at Keimyung University, Daegu, South Korea. His research covers several areas of knot theory, with relations to combinatorics, number theory, and algebra. He earned a PhD from the Free University of Berlin.
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