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

Abstract

For each braid $\beta \in \mathfrak {Br}_n$ 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 ${\mathrm {Hilb}}_{1,n}^{\textit{free}}$. We show that the triply graded vector space of the hypercohomology $\mathbb {H}( \mathbb {S}_{\beta }\otimes \wedge ^\bullet (\mathcal {B}))$ 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 ${\mathrm {Hilb}}_{1,n}\subset {\mathrm {Hilb}}_{1,n}^{\textit{free}}$, that allows us to relate the triply graded knot homology to the sheaves on ${\mathrm {Hilb}}_{1,n}$.