MT 855: Knot homology theories

Spring 2022

Time: Tuesday 12:30-1:50pm and Thursday 9:30-10:50am ET, January 18 - May 5. No class March 8 and 10 (spring break) and April 14 (Easter).

Physical location: Maloney 560

Virtual location: https://bccte.zoom.us/j/5219852726

Lecture Notes.

(just a few) Problem Sets.

Professor: Josh Greene

Course description:

This course will focus on two modern knot invariants, Khovanov homology and knot Floer homology. The first part (~2/3) of the course will introduce each one, compare and contrast them, and relate them to classical topics in knot theory. It will build to two famous applications: using Khovanov homology to prove Milnor's conjecture on the slice genus of torus knots (after Rasmussen), and using knot Floer homology to detect the genus and fiberedness of a knot (after Juhász). The second part (~1/3) of the course will survey some more recent literature surrounding the invariants, with a focus on applications to knot cobordisms and concordance.

The material of MT808 G/T I should suffice as background for the course. The material we will cover in the first part of the course is standard comprehensive exam material for students intending to work around Floer and Khovanov homology. The material in the second part of the course will lead into some possible research topics. I have tried to make the course modular, so students can attend different portions depending on their interest and background. In particular, students who are comfortable with the material of the first part may wish to attend only the second part. I will update this page regularly to announce the material of each forthcoming lecture.

I envision the following breakdown of the material:

Unit 0. (1 lecture) Overview.

Unit 1. (~6 lectures) The Jones polynomial, Khovanov homology, and the slice genus.

Unit 2. (~6 lectures) The Alexander polynomial and Heegaard / knot Floer homology.

Unit 3. (~6 lectures) Sutured Floer homology and genus / fiberedness detection.

Unit 4. (~10 lectures) Recent topics.

Post-mortem:

Units 0-3 spanned the length of the entire course!

Maybe Unit 4 will turn into next Spring's course...

References:

There are many excellent references for the material of the first three units. For now, let me suggest just one or two expository references for each one, which we will closely follow.

Unit 1: Rasmussen, Knots, polynomials, and categorification, Chapters 1 and 3.

Unit 2: Ozsváth and Szabó, An introduction to Heegaard Floer homology.

Also see Manolescu, An introduction to knot Floer homology; Rasmussen, Chapter 2; and Altman, Chapter 2 (below).

Unit 3: Altman, Introduction to sutured Floer homology, Chapters 2-4, and Lipshitz, Heegaard Floer homologies.

For the second (non-existent) part of the course (Unit 4), we will examine a selection of articles drawn from (around) this list:

Alishahi, The Bar-Natan homology and unknotting number.

Ballinger, Concordance invariants from the E(-1) spectral sequence on Khovanov homology.

Ballinger, A family of concordance invariants from Khovanov homology.

Livingston, Notes on the knot concordance invariant Upsilon.

Juhász, Miller, and Zemke, Knot cobordisms, bridge index, and torsion in Floer homology.

Levine and Zemke, Khovanov homology and ribbon concordances.

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