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

Alexei Oblomkov, Lev Rozansky

• For each braid β∈Brn we construct a 2-periodic complex Sβ of quasi-coherent C∗×C∗ -equivariant sheaves on the non-commutative nested Hilbert scheme Hilbfree1,n . We show that the triply graded vector space of the hypercohomology H(Sβ⊗∧∙(B)) with B being tautological vector bundle, is an isotopy invariant of the knot obtained by the closure of β . We also show that the support of cohomology of the complex Sβ is supported on the ordinary nested Hilbert scheme Hilb1,n⊂Hilbfree1,n , that allows us to relate the triply graded knot homology to the sheaves on Hilb1,n .