EVENT DETAILS AND ABSTRACT

Title: Virtual links and algebraic categorification of quantum polynomial invariants
Speaker: Lev Rozansky
Speaker Info: UNC at Chapel Hill
Brief Description:
Special Note: Special Session
Abstract:

Quantum polynomial invariants of links in S^3, such as the Jones, HOMFLY and Kauffman polynomials, were invented about 20-25 years ago, but their topological nature remains largely mysterious. Their rigorous definition is combinatorial, while their conceptual definition in terms of Chern-Simons-Witten path integral is mathematically non-rigorous.

Categorification program, whose pratcital implementation was pioneered by M. Khovanov, suggests to interpret the quantum polynomial invariants as graded Euler characteristics of special chain complexes of graded modules (or vector spaces), associated to links up to homotopy. For the Jones, HOMFLY and Kauffman polynomials, these complexes can be constructed combinatorially with the help of commutative algebra tools.

Virtual liks were invented by L. Kauffman (and independently by M. Polyak and O. Viro) and they represent links in `thick' surfaces. Some quantum polynomial invariants can be extended to virtual links, but the virtual crossings play a minor role in these extensions.

It turns out that virtual crossings play a central role in commutative algebra categorification of quantum polynomials. The ordinary crossings appear to be a certain `homological' deformation of the virtual ones. I will explain this construction at the example of the categorification of the 2-variable HOMFLY polynomial.

The talk is based on my joint work with M. Khovanov (math.QA/0505056, math.QA/0701333).