Diagram Genus, Generators, And Applications (chapman & Hall/crc Monographs And Research Notes In Mathematics)

In knot theory, diagrams of a given canonical genus can be described by means of a finite number of patterns ('generators'). This text presents a self-contained account of the canonical genus: the genus of knot diagrams. 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 Hirasawa's 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 Front Cover 1 Contents 7 Preface 11 List of Figures 15 List of Tables 17 Symbol Description 19 Chapter 1 - Introduction 21 Chapter 2 - Preliminaries 33 Chapter 3 - The maximal number of generator crossings and ∼-equivalence classes 67 Chapter 4 - Generators of genus 4 83 Chapter 5 - Unknot diagrams, non-trivial polynomials and achiral knots 89 Chapter 6 - The signature 119 Chapter 7 - Braid index of alternating knots 123 Chapter 8 - Minimal string Bennequin surfaces 139 Chapter 9 - The Alexander polynomial of alternating knots 147 Chapter 10 - Outlook 165 Bibliography 179 Glossary 187 Back Cover 191