Chapman and Hall/CRC
174 pages | 17 B/W Illus.
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.
"Diagram Genus, Generators and Applications contains a systematical study of combinatorial properties of knot diagrams. It focuses on diagrams that represent the canonical genus of a knot, i.e., the minimal genus of all Seifert surfaces for a given knot that are obtained by applying Seifert’s algorithm to diagrams of the knot. The book contains the complete classification of knots up to canonical genus 4. This classification has lots of applications … The book … will certainly become a reference in this area. It is very clearly written and contains enough background material so that it can be used by graduate-level students to learn the subject and do work in this area on their own."
—Thomas Fiedler, Institut de Mathématiques, Université Paul Sabatier, Toulouse
"This book provides an essential resource for anyone currently doing research or interested in doing research on surfaces in knot complements and their applications. Enough background is included so non-experts can follow the exposition and appreciate the myriad results that ensue."
—Professor Colin Adams, Williams College
"This monograph is a systematic account of combinatorial knot theory, with a particular focus on spanning surfaces arising from Seifert’s construction. It includes a brief and nicely written introduction to knot theory, concentrating on the background needed for a diagrammatic treatment of knots, including the range of classical and modern knot polynomials.
A strong feature of this book, and indeed much of the author’s work elsewhere, is the identification of diagrammatic examples with awkward or unexpected properties, and an analysis of the techniques that can be used effectively on them. This can provide examples that can’t possibly be tackled by certain procedures, and thus directs attention to places where the current repertoire of techniques is lacking.
The main topic developed is the notion of diagram genus, or canonical genus, based on Seifert’s algorithm. The related graph theory leads to the selection of a class of alternating knot diagrams, termed generators, and a substantial account of these up to genus 4 is given.
This is followed by the discussion of a number of combinatorial results and conjectures. In particular, some nice results for alternating or positive knots are given and their possible extension to the case when k of the knot crossings are switched is explored for small values of k. The earlier calculations are used to extend the knowledge of these results to cover knots with fewer restrictions on their genus or crossing number.
There is a good account of the combinatorics for recognizing when a knot diagram actually represents the trivial knot. It is surprisingly easy to draw diagrams of the trivial knot with relatively few crossings that do not have an immediately obvious simplification, and some examples are included in the illustrations.
A further section covers the question of finding the braid index for an alternating knot, and the conditions under which the Morton-Franks-Williams bound turn out to be sharp. The concluding section is intended as an appetizer for others and includes a variety of annotated questions and conjectures.
The carefully written text is aimed at a graduate-level readership. It gives a comprehensive view of combinatorial questions, both in the monograph itself and in the well-annotated bibliography, and would serve both well as a reference and a source of new ideas.
—H.R. Morton, Department of Mathematical Sciences, University of Liverpool
The beginning of knot theory
Reidemeister moves and invariants
Combinatorial knot theory
Genera of knots
Overview of results
Issues of presentation
Knots and diagrams
Crossing number and writhe
Knotation and not-tables
Seifert surfaces and genera
Braids and braid representations
MWF inequality, Seifert graph, and graph index
Knots vs. links
The maximal number of generator crossings and ~-equivalence classes
Generator crossing number inequalities
An algorithm for special diagrams
Proof of the inequalities
Applications and improvements
Generators of genus 4
Unknot diagrams, non-trivial polynomials, and achiral knots
Some preparations and special cases
Reduction of unknot diagrams
Non-triviality of skein and Jones polynomial
On the number of unknotting Reidemeister moves
Achiral knot classification
Braid index of alternating knots
Motivation and history
Hidden Seifert circle problem
Modifying the index
Minimal string Bennequin surfaces
Statement of result
The restricted index
Finding a minimal string Bennequin surface
The Alexander polynomial of alternating knots
The log-concavity conjecture
Complete linear relations by degree
Legendrian invariants and braid index
Minimal genus and fibering of canonical surfaces
Wicks forms, markings, and enumeration of alternating knots by genus
Canonical genus bounds hyperbolic volume
The relation between volume and the slN polynomial
Everywhere equivalent links