# Khovanov Homology in Austin

The main page for the UT Austin Khovanov Homology Workshop is on Carl's webpage; you should go there for organisational information. Lecture notes are now available here.

This page is to help people prepare for their talks. It's editable by anyone -- jump right in!

## Contents

# Talks

## Jones-Reshetikhin-Turaev-Chern-Simons-Witten Theory. (Thursday)

Before we categorify, we should have a good idea what we are categorifying.

### Talk 1. Jones polynomial and Reshetikhin-Turaev invariants of knots. (9:00-9:45, 10:00-10:45, Parker Lowrey)

Ref: Bakalov-Kirilov, Tensor categories and modular functors, chapter 2.

### Talk 2. Reshetikhin-Turaev invariants of 3-manifolds. (11:00-11:45, 14:00-14:45, Deepee Khosla)

ref: Bakalov-Kirilov, chapter 3.

### Talk 3. The Chern-Simons TQFT. (15:00-15:45, 16:00-16:45, Alex Kahle)

Iâ€™m not sure what the best reference is. Perhaps it is best to talk to Dan Freed.

For the first two of these talks, you might also want to look at Kevin Walker's unpublished notes, at http://canyon23.net/math

## Basic definition of Khovanov homology. (Friday)

Now we are ready for it.

### Talk 4. Khovanov homology. (9:00-9:45, 10:00-10:45, Jonathan Williams and Sean Bowman)

Ref: Bar-Natan, On Khovanovâ€™s categorification of the Jones polynomial, Khovanov, A functor valued invariant of tangles, Khovanov, An invariant of tangle cobordisms.

### Talk 5. Bar-Natanâ€™s version of Khovanov homology. (11:00-11:45, 14:00-14:45, Scott Morrison)

Ref: Bar-Natan, Khovanovâ€™s homology for tangles and cobordisms.

Here are some suggestions of things that make be worth hitting, if there's time: --Scott 21:01, 5 December 2006 (PST)

- the delooping isomorphism.
- simplifying complexes (for these two, see Fast Khovanov Homology Computations, Dror Bar-Natan).
- recovering Khovanov's bimodule invariant for tangles from Bar-Natan's construction (ask Scott).

### Talk 6. Rasmussenâ€™s proof of the Milnor conjecture. (15:00-15:45, 16:00-16:45, Jeremy Van Horn)

Ref: Rasmussen, Khovanov homology and the slice genus.

This is a purely topological application of Khovanov homology.

This requires introducing Lee homology; you might look at The Karoubi Envelope and Lee's Degeneration of Khovanov Homology, Bar-Natan and Morrison. It would be great to explain Lee homology (and Rasmussen's genus bounds) in the `polynomials in a triple torus' theory; perhaps ask me about this. --Scott 21:01, 5 December 2006 (PST)

## Categorification using category O. (Saturday)

One approach to categorifying the Jones polynomial and other aspects of the representation theory of Uq(sl2) using parabolic blocks of category O for gln.

### Talk 7. Categorification via category O. (9:00-9:45, 10:00-10:45, Carl Mautner)

Ref: Bernstein-Frenkel-Khovanov, A categorification of the Temperley- Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors and Stroppel, Categorification of the Temperley-Lieb algebra, Tan- gles and cobordisms via projective functors. Recently, Brundan-Kleshchev showed how this categorification fits into a more general pattern which categorifies all polynomial representation of gln.

### Talk 8. Categorification of polynomial representation of gln. (15:00-15:45, 16:00-16:45, Joel Kamnitzer)

Ref: Brundan-Kleshchev, Representations of shifted Yangians and finite W-algebras.

## Symplectic geometry knot invariants. (Sunday)

Seidel-Smith have constructed a knot invariant which they conjecture to equal Khovanov homology (after collapsing the bigrading).

### Talk 9. Seidel-Smith knot homology (9:00-9:45, 10:00-10:45, Maksim Maydanskiy)

Ref: Seidel-Smith, A link invariant from the symplectic geometry of nilpo- tent slices, Manolescu, Nilpotent slices, Hilbert schemes, and the Jones poly- nomial.

## More general Khovanov homology. (Sunday)

Khovanov has introduced a few approaches towards understanding Khovanov homology corresponding to other Lie algebras.

### Talk 10. Foams and sl(3) Khovanov homology. (11:00-11:45, 14:00-14:45, Scott Morrison)

Ref: Khovanov, sl(3) knot homology.

### Talk 11. Khovanov-Rozansky homology. (15:00-15:45, 16:00-16:45, Eric Katerman)

Ref: Khovanov-Rozansky, Matrix factorizations and link homology.

This talk potentially covers a lot of ground! It would be great to explain the 'uniform' construction of the SU(n) and HOMFLY invariants -- look up Rasmussen's recent papers. --Scott 21:06, 5 December 2006 (PST)

# References

- B. Bakalov, A. A. Kirillov; Lectures on Tensor Categories and Modular Functors, Chap. 2-3. (You can take a look at Kevin Walker's notes while you're at it.)
- D. Bar-Natan; On Khovanov's categorification of the Jones polynomial.
- M. Khovanov; An invariant of tangle cobordisms.
- D. Bar-Natan; Khovanov's homology for tangles and cobordisms.
- Rasmussen; Khovanov homology and the slice genus.
- J. Bernstein, I. Frenkel, M. Khovanov; A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors.
- C. Stroppel; Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors.
- J. Brundan, A. Kleshchev; Represenations of shifted Yangians and finite W-algebras.
- P. Seidel, I. Smith; A link invariant from the symplectic geometry of nilpotent slices.
- I. Smith; Monodromy, vanishing cycles, knots and the adjoint quotient.
- C. Manolescu; Nilpotent slices, Hilbert schemes, and the Jones polynomial.
- C. Manolescu; Link homology theories from symplectic geometry.
- M. Khovanov; sl(3) knot homology.
- M. Khovanov, L. Rozansky; Matrix factorizations and link homology.