Title: Soergel bimodules and the categorification of the 2-variable HOMFLY-PT polynomial
Speaker: Professor Lev Rozansky
Speaker Info: University of North Carolina
This talk is based on our joint work with M. Khovanov.Date: Thursday, May 18, 2006
The 2-variable HOMFLY-PT polynomial is a topological invariant of oriented links in S^3. Its definition is purely combinatorial and its nature is mysterious. To a diagram of a link we associate a chain complex of bi-graded modules, whose homotopy class is a topological invariant of the link and whose bi-graded Euler characteristic equals the HOMFLY-PT polynomial of the link. Our construction is based on a representation of the braid group by endo-functors of a certain category, which has been described previously in the works of W. Soergel and R. Rouquier. The endo-functors are represented by bi-modules and we close the braid into a link by taking their Hochschild homology. The whole construction is based on elementary manipulations with polynomial algebra.
The HOMFLY-PT polynomial has an important specialization: if we set t=q^N, then the resulting 1-variable polynomial is related to the representation theory of the quantum group SU_q(N). This polynomial admits a similar categorification, if we modify the polynomial algebra underlying the Soergel's construction, and replace the Koszul complexes with Koszul matrix factorizations.