Diagram Genus, Generators, and Applications

Alexander Stoimenow

Sept 2018 · CRC Press

About this eBook

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.

About the author

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.