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

2016/08/10 by Oblomkov, Alexei, Rozansky, Lev · 3 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1608.03227

Abstract

For each braid β∈ Br_n we construct a 2-periodic complex \mathbbS_β of quasi-coherent ℂ^*× 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 ℍ( \mathbbS_β⊗ \wedge^\bullet (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 \mathbbS_β is supported on the ordinary nested Hilbert scheme Hilb_1,n⊂ Hilb_1,n^free, that allows us to relate the triply graded knot homology to the sheaves on Hilb_1,n.

Cited by

• Doubly-graded exponential growth of colored torus knot homology
• A Lie-theoretic generalization of some Hilbert schemes
• Potent categorical representations

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