Modern research methods in knot theory can be more-or-less grouped into several categories:
Introductory Lectures on Knot Theory contains 18 survey papers that cover most of the above categories. Knot theorists of any persuasion are likely to find essays of interest in this volume, although the diagrammatic approach is the best represented, and the geometric approach is the least well represented. The essays vary tremendously in their general accessibility and in their presumed audience. Some of the essays are very well written, but others could have used more editing. The essays in the volume are in alphabetical order by author last name; organizing the essays by topic would have made for a more coherent volume. Readers interested in just one or two of the essays should note that a few of them are available for free on the arXiv. Below, I’ll highlight a some of the essays that I found most interesting. I’ve grouped the papers by their topic and methods.
“Quandle coloring” is one generalization of 3-coloring. Przytycki’s paper doesn’t discuss quandles much, but they are covered in Carter’s paper “A survey of quandle ideas”. Carter is mostly concerned with the history of quandles. His account provides useful attributions of the fundamental ideas. Carter’s paper is more concerned with the algebra of quandles than with their connection to topology, but he does discuss some topological applications of the concepts. Fenn’s paper “Finding knot invariants from diagram coloring” shows how to use quandles and similar algebraic objects to obtain invariants such as the Alexander polynomial of a knot. Fenn’s paper is a good source for basic examples and applications of quandles. His paper does not provide many proofs, but does contain an extensive bibliography.
The other essays taking a diagrammatic approach are by Chmutov, Ilyutko-Manturov, Jablan-Sazdanović, Manturov, Kauffman, and Lambropolou. Of these, Kauffmann’s introduction to Khovanov homology is likely to be the one of interest to most people. It is based on Bar-Natan’s influential account of Khovanov homology, and is a helpful companion to Bar-Natan’s work.
[D] Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, Bert Wiest. Why are braids orderable? Panoramas et Synthèses 14. Société Mathématique de France, 2002.
Scott Taylor is a knot theorist at Colby College.