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

Abstract

For each braid $\beta \in B{r}_n$ we construct a $2$-periodic complex ${\mathbb{S}}_{\beta }$ of quasi-coherent ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$-equivariant sheaves on the non-commutative nested Hilbert scheme $Hil{b}_{1,n}^{free}$. 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 $Hil{b}_{1,n}\subset Hil{b}_{1,n}^{free}$, that allows us to relate the triply graded knot homology to the sheaves on $Hil{b}_{1,n}$.

arXiv commentMany misprints are corrected, some proofs are expanded. To appear in Selecta Math. Comments are welcome!