## This is an old course webpage

## 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.

**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