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Math 285F: Khovanov homology

Spring 2022

Instructor: Sucharit Sarkar.
Class: M 13:00-14:50, F 14:00-14:50, MS 6201.
Office hours: By appointment, MS 6909.

Syllabus: We will study Khovanov homology, which is a very modern invariant of knots and a categorification of the famous Jones polynomial. Jones polynomial was one of the first invariants of knots which was not geometrically defined and its precise geometric meaning is still a mystery. Khovanov homology lifts the Jones polynomial one level higher, and discovers surprising connections between representation theory and knot theory. We will also study some applications of these invariants, such as Kauffman's proof of Tait's conjecture about alternating knots or Rasmussen's alternate proof of Milnor's conjecture, first proved by Kronheimer-Mrowka, about torus knots. You should be familiar with the basics of topology and algebra. Anything else that we need, we will cover them in class; so this can be your first course in low-dimensional topology, and can be a good learning opportunity for many modern mathematical techniques. Since this a very new subject, there are no good books written on the topic, so the lectures will follow papers instead. Here is a tentative list of papers (a subset of) which we plan to cover.
Paper Description
A categorification of the Jones polynomial The original paper by Khovanov.
On Khovanov's categorification of the Jones polynomial A very user-friendly introduction to Khovanov homology.
Patterns in knot cohomology I Reduced Khovanov homology.
An endomorphism of the Khovanov invariant Lee's deformation of Khovanov homology.
Link homology and Frobenius extensions Khovanov homology with respect to other Frobenius algebras.
Khovanov homology and the slice genus Rasmussen's computation of 4-ball genus of torus knots.
Khovanov's homology for tangles and cobordisms Bar-Natan's tangle invariant.
The universal Khovanov link homology theory Information content of Bar-Natan's tangle invariant.
A functor-valued invariant of tangles Khovanov's tangle invariant.
An invariant of link cobordisms from Khovanov homology Knot cobordism maps in Khovanov homology.
A Khovanov homotopy type A Khovanov stable homotopy type.
Khovanov homotopy type, Burnside category, and products Another construction of Khovanov stable homotopy type.
A Steenrod square on Khovanov homology Stable cohomological operations on Khovanov homology.
A refinement of Rasmussen's s-invariant New invariants from the stable homotopy type.
Matrix factorizations and link homology Khovanov-Rozansky's sl(n) homologies.
Matrix factorizations and link homology II Khovanov-Rozansky's HOMFLY-PT homology.
Some differentials on Khovanov-Rozansky homology Spectral sequences connecting the sl(n) homologies.


Grading: Since this is a graduate topics course, there is no grading per se. However, for the undergraduate students attending the course and who will need a grade, their grade will be entirely based on attendance and HW which is due at the end of the quarter. The HW is the PDF below (which will be updated after every lecture) and involves working out the details that I skipped in class.
HW