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

Alexei Oblomkov ^[2] ; Lev Rozansky ^[1]
1. [1] University of North Carolina at Chapel Hill
  
  University of North Carolina at Chapel Hill
  
  Township of Chapel Hill, Estados Unidos
2. [2] University of Massachusetts at Amherst
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 3, 2018, págs. 2351-2454
Idioma: inglés
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Resumen
- 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 .