Introductory Lectures in Knot Theory

Modern research methods in knot theory can be more-or-less grouped into several categories:

• Cut-and-paste: these methods rely on detailed analyses of the intersection between various surfaces living in 3-dimensional manifolds. Of all the methods, these are most directly connected to the topology of the knot. One of the triumphs of these methods was the proof by Gordon and Luecke that knots in the 3-sphere are determined by their complements.
• Diagrammatic: these methods use knot diagrams to create numbers, polynomials, or other arithmetic or algebraic structures that are invariant under the moves that relate two diagrams of the same knot. One of the jewels of diagrammatic knot theory is the new proof by Rasmussen using Khovanov homology of the so-called “Milnor conjecture” concerning the slice genus of torus knots (originally proved by Kronheimer-Mrowka using Donaldson theory).
• Geometric: In the 1980s, Thurston showed that the complement of “most” knots can be given a unique geometric structure modelled on the geometry of hyperbolic 3-space. The hyperbolic structure can be used to give a thorough understanding of the geometry and topology of the knot complement. Invariants arising from the hyperbolic structure (such as volume) are among the most powerful knot invariants.
• Analytic: In the past two decades, Kronheimer, Mrowka, Oszváth, Szabó, and others have applied powerful analytic methods originating in 4-manifold theory to knots in 3-manifolds to obtain some stunning results. In a somewhat different direction, quantum knot invariants, based on work of Witten, have also become an important research topic in knot theory. A lot of recent work has made connections between these analytic methods and the diagrammatic and geometric properties of knots.
• Numerical: In an effort to effectively apply knot theory to the study of real-world problems arising from engineering and biology, several researchers have explored “physical knot theory”: a version of knot theory where knots have length and thickness. There are not many theorems, but there has been a lot of work using numerical methods to study the “space” of physical knots.

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.

• Kauffman and Lambropoulou’s essay “Hard unknots and collapsing tangles” is a very readable introduction to diagrammatic knot theory. It is concerned with the problem of creating diagrams of the unknot that require many Reidemeister moves to unknot. Along the way, it provides a very accessible introduction to Conway’s theory of rational tangles. A few digressions highlight connections to other parts of mathematics, most notably hyperbolic geometry.
• The essay “Derivation and interpretation of the Gauss linking number” by Ricca and Nipoti, gives an interesting account of Gauss’ linking number and its applications to physics in the 19th century. Gauss himself never wrote much about knots, and only mentions the linking number briefly in his diary. The concept, however, is “fundamental in the subsequent development of knot theory, general topolgy and modern topological field theory.” The authors give a possible route by which Gauss may have discovered the linking number. Their article should be readable by anyone with some knowledge of vector calculus and a little bit of topology and knot theory.
• Przytycki’s essay “The Trieste look at knot theory” is probably the next most accessible paper in the volume: it begins with knots in Mesopotamia and ends with an account of recent work on coloring knots. The ability or inability of a knot diagram to be 3-colored is a simple diagrammatic knot invariant and is the probably the knot invariant most people are introduced to first. This idea has been developed considerably since its discovery by Ralph Fox in the 1950s.

  “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.

• Millet’s and Rawdon’s essays are both introductions to physical knot theory. Both are very well-written and are good essays to hand to people interested in applications of knot theory. Millet’s essay is more of an overview, while Rawdon’s essay is mostly about generating models of physical knots and measuring associated geometric and topological properties.
• Gordon’s essay “Exceptional Dehn filling” is a survey of results concerning the question: “What Dehn surgeries on what hyperbolic links produce non-hyperbolic 3-manifolds?” The paper surveys current knowledge on the topic, stating theorems and providing references to examples and proofs. Many of the results mentioned rely on cut-and-paste techniques, but hyperbolic geometry also plays an important role. One interesting contribution of this paper is its exposition of the notion of “ancestral finiteness” for exceptional surgeries. Gordon raises some interesting questions concerning exceptional surgeries in general, and ancestral finiteness in particular.
• “Braid order, sets, and knots” by Dehornoy is a provocative account of the “set-theoretical roots of the D[ehornoy]-ordering” on the braid groups. The D-ordering is a very special left-invariant total order on the braid groups. The Math Reviews review of [D] calls the discovery of the D-ordering the “the most significant result of the past ten years in the study of the braid groups.” It turns out, as explained in Dehornoy’s article, that the discovery of this ordering has its roots in the large cardinal axioms in set theory. Dehornoy compares the situation to the relationship between physics and mathematics where “physical assumptions [are used] to guess some statement that is subsequently passed to the mathematician for a formal rigorous proof. The situation here is quite similar: using logical assumptions …, one guesses some statement …, and then passes it to the mathematician for … a proof that involves no extra logical axiom.” I find this a very exciting statement and would love to know of more examples of this kind of interplay between logic and geometric topology.
• The essays by Beliakova-Le, Morton, and Pichai are all concerned with quantum invariants of knots. Beliakova and Le survey constructions of so-called “unified invariants”. These invariants attempt to explain the relationship between the quantum link invariants of Witten, Reshetikhin-Turaev and the 3-manifold invariants of Le-Murakami-Ohtsuki. The article is quite technical. Morton’s article is a nice introduction to the bracket approach to quantum invariants of knots and 3-manifolds. Pichai’s essay explains how Chern-Simons gauge theory can be used to detect the chirality of knots and explains why it cannot distinguish all mutants.

Reference:

[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.