Diagram Genus, Generators, and Applications

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 Mathematiques, Universite 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. Features * 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