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

TL;DR

This paper constructs a new sheaf-theoretic framework on Hilbert schemes to produce knot invariants, linking knot homology with algebraic geometry and sheaf theory.

Contribution

It introduces a novel complex of sheaves on non-commutative Hilbert schemes that yields triply graded knot invariants and relates them to classical Hilbert schemes.

Findings

The triply graded vector space of hypercohomology is an isotopy invariant of knots.

Supports of cohomology are contained in the classical nested Hilbert scheme.

The framework connects knot homology with sheaves on Hilbert schemes.

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 $Hilb_{1,n}$.