Diagram Genus, Generators, and Applications

1st Edition

Alexander Stoimenow

Chapman and Hall/CRC
Published February 9, 2016
Reference - 174 Pages - 17 B/W Illustrations
ISBN 9781498733809 - CAT# K26329
Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Summary

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.