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

Research paper by Alexei Oblomkov, Lev Rozansky

Indexed on: 01 Feb '18Published on: 31 Jan '18Published in: Selecta Mathematica

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 $${\mathrm {Hilb}}_{1,n}$$.