Knot theory

An introduction to knot theory

The course is held in Salle Hypatia, on Mondays 14–17.

We will cover material from Lickorish's An introduction to knot theory (L) and Roflsen's Knots and links (R).

This is a diary of the course:

Lecture 1 (14/01): definitions of knots, knot equivalence and isotopy, knot diagrams, connected sum, mirrors and reverses; the unknot and torus knots; knot invariants, regular neighbourhoods, the knot complement and its homology, the knot group, the Wirtinger presentation; proof that the trefoil is not unknotted. (Roughly [R; Chapters 2E, 3A–D].)

Lecture 2 (21/01): genus, genus characterisation of the unknot, Seifert's algorithm, additivity of the genus, indecomposable and prime knots, structure theorem for the set of knots, non-cancellation theorems; Reidemeister moves, diagrammatic invariants. (Roughly [L; Chapter 2] and [R; Chapter 5A].)

Lecture 3 (28/01): linking number, infinite cyclic covers, Seifert matrix, definition of the Alexander module and the Alexander polynomial. (Roughly [L; Chapters 6-7] and [R; Chapters 7A, 8C].)

Lecture 4 (04/02): the infinite cyclic cover of a knot complement, relationship with Seifert matrices; fibred knots, their Alexander polynomials; torus knots are fibred. (Roughly [L; Chapters 6-7] and [R; Chapters 7A, 8C, 10H].)

Lecture 5 (11/02): the Alexander–Conway polynomial, skein relation; the Levine–Tristram signature; concordances and the concordance group, the Fox–Milnor theorem, and slice genus bounds from the signature. ([L; Chapter 8] and [R; Chapters 8E-F].)

Lecture 6 (04/03): alternating knots, their history and recent development; Tait's conjectures; the Kauffman bracket, the Jones polynomial, and their properties; invariance under mutation; history of the Jones polynomial. ([L; Chapter 3]).

Lecture 7 (11/03): state-sum presentation of the Kauffman bracket and the proofs of the first two Tait conjectures; definition and examples of the Arf invariant; Alexander, Arf, and Jones. ([L, Chapters 5 and 10].)

Lecture 8 (18/03): smooth exotica, low dimensions vs high dimensions; existence of an exotic R^4; Khovanov homology, Lee theory, Rasmussen's s-invariant, the Milnor conjecture, non-sliceness of the Whitehead double of the trefoil.

The homework problems are here: sheet 1, sheet 2, sheet 3. The seminar topics are here; you are welcome to suggest your own topic, if you prefer (pending my approval).