Geometry and Topology Seminar: "Khovanov-Rozansky homology and sheaves on Hilbert scheme of points on the plane."

Abstract: Talk is based on the joint work with Lev Rozansky. I will explain a construction that attaches to a $n$-stranded braid $\beta$ a two-periodic complex $S_\beta$ of $\mathbb{C}^*\times \mathbb{C}^*$-equariant sheaves on $Hilb_n(\mathbb{C}^2)$ such that the $H^*(S_\beta)$ is a categorification of the Oceanu-Jones trace. We show the corresponding link homology coinside with the tripy graded Khovanov-Rozansky link homology which categorifies of Jones construction of HOMFLYPT polynomial. We also show that $S_{\beta FT}=S_\beta \otimes L$ where $FT$ is a full twist and $L$ is a generator of the Picard group of $Hilb_n(\mathbb{C}^2)$. The natural invoultion of $\mathcal{C}^2$ results in Poincare duality of the Khovanov-Rozansky homology (conjectured in 2005). As an application we obtain explicit an combinatorial formula for  Khovanov-Rozansky homology of torus knots.

Host: Aliakbar Daemi