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

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 \({\mathrm {Hilb}}_{1,n}\).