1st Edition

Diagram Genus, Generators, and Applications

By Alexander Stoimenow Copyright 2016
    192 Pages 17 B/W Illustrations
    by Chapman & Hall

    192 Pages 17 B/W Illustrations
    by Chapman & Hall

    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.

    The beginning of knot theory
    Reidemeister moves and invariants
    Combinatorial knot theory
    Genera of knots
    Overview of results
    Issues of presentation
    Further applications

    Knots and diagrams
    Crossing number and writhe
    Knotation and not-tables
    Seifert surfaces and genera
    Diagrammatic moves
    Braids and braid representations
    Link polynomials
    MWF inequality, Seifert graph, and graph index
    The signature
    Genus generators
    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

    The signature

    Braid index of alternating knots
    Motivation and history
    Hidden Seifert circle problem
    Modifying the index
    Simplified regularization
    A conjecture

    Minimal string Bennequin surfaces
    Statement of result
    The restricted index
    Finding a minimal string Bennequin surface

    The Alexander polynomial of alternating knots
    Hoste’s conjecture
    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
    Crossing numbers
    Canonical genus bounds hyperbolic volume
    The relation between volume and the slN polynomial
    Everywhere equivalent links


    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.

    "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.


    • A comprehensive account of diagram-centered results in knot theory
    • Focus on Seifert’s construction of oriented-spanning surfaces
    • Analysis of diagrams representing the unknot and their reduction by Reidemeister moves
    • Careful and persuasive writing
    • An excellent reference text and source of ideas"

    —H.R. Morton, Department of Mathematical Sciences, University of Liverpool