Knot Homology and sheaves on the Hilbert scheme of points on the plane

Abstract

For each braid $\beta\in Br_n$ we construct a $2$-periodic complex $\mathbb{S}_\beta$ of quasi-coherent $\mathbb{C}^*\times \mathbb{C}^*$-equivariant sheaves on the non-commutative nested Hilbert scheme $Hilb_{1,n}^{free}$. We show that the triply graded vector space of the hypecohomology $ \mathbb{H}( \mathbb{S}_{\beta}\otimes \wedge^\bullet (\mathcal{B}))$ with $\mathcal{B}$ being tautological vector bundle, is an isotopy invariant of the knot obtained by the closure of $\beta$. We also show that the support of cohomology of the complex $\mathbb{S}_\beta$ is supported on the ordinary nested Hilbert scheme $Hilb_{1,n}\subset Hilb_{1,n}^{free}$, that allows us to relate the triply graded knot homology to the sheaves on $Hilb_{1,n}$.