Lectures

Notes graciously provided by Nic Petit.

Recommended references follow each lecture.

Lecture 1 (1/18): Overview. Knots and their properties, polynomial invariants, knot homology theories, and some applications.

Part 1. Khovanov homology.

Lecture 2 (1/20): The Jones polynomial, Kauffman's state sum model, the crossing number, and Tait's conjecture.

Lickorish, An Introduction to knot theory, Chapters 3 and 5.

It is based on

Kauffman, State models and the Jones polynomial.

We also borrowed from

Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links.

Lecture 3 (1/25): TQFT and the construction of Khovanov homology.

Rasmussen, Knots, polynomials, and categorification., Sections 3.1-3.4

You can also skim the early bits of

Bar-Natan, On Khovanov's categorification of the Jones polynomial.

Lecture 4 (1/27): Gradings on Khovanov homology, the Jones polynomial, and a few words about invariance.

Rasmussen, Section 3.5-3.6

Bar-Natan, ibid.

Lecture 5 (2/1): Invariance, naturality, and functoriality.

Rasmussen, Section 3.7.

Skim the introduction of

Juhász, Thurston, Zemke. Naturality and mapping class groups in Heegaard Floer homology.

Also see

Jacobsson, An invariant of link cobordisms from Khovanov homology.

Lecture 6 (2/3): Distinguishing slice disks using Khovanov homology; Seifert's algorithm.

Hayden and Sundberg, Khovanov homology and exotic surfaces in the 4-ball.

Almost any introductory text on knot theory describes Seifert's algorithm. For instance, see

Rasmussen, Section 2.6

Lecture 7 (2/8): The skein exact sequence; signature and determinant; Khovanov homology of alternating knots.

Manolescu and Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links,

Rasmussen, Knot polynomials and knot homologies, Sections 4.1-4.4. (This is not the same as the usual Rasmussen reference!)

Lecture 8 (2/10): Deformations.

Rasmussen (usual Rasmussen reference), start of Section 3.8.

Lecture 9 (2/15): Deformations, filtrations, and the s-invariant.

Lipshitz-Sarkar, A mixed invariant of non-orientable surfaces in equivariant Khovanov homology, Prop 2.1.

Turner, Khovanov homology and diagonalizable Frobenius algebras.

Rasmussen, Section 3.8.

Lecture 10 (2/17): The Milnor conjecture.

Rasmussen, Section 3.8.

Rasmussen, Khovanov homology and the slice genus, Section 5.2.

Lecture 11 (2/22): Filtered complexes and spectral sequences.

Baldwin, On the spectral sequence from Khovanov homology to Heegaard Floer homology, Section 4.

Bar-Natan, On Khovanov's categorification of the Jones polynomial, Section 3.6.

Manolescu and Marengon, The knight moves conjecture is false.

Part 2. Heegaard and knot Floer homology.

Lecture 12 (2/24): Heegaard diagrams.

Hom, Lecture notes on Heegaard Floer homology, through Section 1.2.

Ozsváth and Szabó, An introduction to Heegaard Floer homology, through Section 3.

Szabó, Lecture notes on Heegaard Floer homology, Sections 1-3.

Lecture 13 (3/1): Morse theory and Heegaard diagrams.

Ozsváth and Szabó, Section 3 and the start of Section 10.

Lecture 14 (3/3): Doubly-pointed Heegaard diagrams; first comparison of HFK and Kh.

Ozsváth and Szabó, Section 12.

*** Spring Break ***

Lecture 15 (3/15): The Heegaard Floer chain complex, part 1.

Ozsváth and Szabó, Sections 4-5.1.

Lecture 16 (3/17): The Heegaard Floer chain complex part 2.

Ozsváth and Szabó, Section 8.

Lecture 17 (3/22): Morse homology.

Audin and Damian, Morse theory and Floer homology. Skim Chapters 2 and 3.

Lecture 18 (3/24): Lagrangian Floer homology.

Audin and Damian, skim Sections 6.1 and 6.2. Section 6.2 gives a nice overview of the idea of Hamiltonian Floer homology.

Lecture 19 (3/29): First computation of HF. Spin^c structures and generators.

Ozsváth and Szabó, Section 6.

Lecture 20 (3/31): More about Spin^c structures.

Ozsváth and Szabó, Section 6.

Lecture 21 (4/5): Structure of HF. Gradings and domains.

Ozsváth and Szabó, Section 5.2.

Lecture 22 (4/7): The Maslov grading.

Ozsváth and Szabó.

Lecture 23 (4/12): Positivity of intersections and the knot filtration.

Ozsváth and Szabó, Sections 7 and 10.

Milnor and Stasheff, Characteristic Classes, Chapter 13 and the beginning of Chapter 14.

*** Holiday ***

Lecture 24 (4/21): Knot Floer homology.

Ozsváth and Szabó, Sections 11-13.

Part 3. Sutured Floer homology.

Lecture 25 (4/26): Fibering the figure-eight knot complement and sutured manifolds (part 1).

Gabai, Detecting fibred links in S^3, Sections 1-2 (skim the rest).

Gabai, Foliations and the genera of links (skim).

*** ICERM Worskhop ***

Lecture 26 (5/3): Fibering the figure-eight knot complement and sutured manifolds (part 2), and taut foliations.

Skim Gabai's papers.

Lipshitz, Heegaard Floer homologies lecture notes, Section 2.

Lecture 27 (5/5): Sutured manifolds and SFH.

Altman, Introduction to sutured Floer homology. Section 4, pp.39-42.

Lipshitz, Heegaard Floer homologies lecture notes, Section 2 and beginning of Section 3.

Juhász, A survey of Heegaard Floer homology, Section 6.

Lecture 28 (5/10): SFH.

Altman, skim Sections 4.2-4.5.

Lipshitz, Section 3.

Juhász, Section 6.

Lecture 29 (5/12): HFK detects genus and fiberedness.

Altman, Sections 4.6-4.7.

Lipshitz, Section 4.

Juhász, Section 6.