Volume 26, issue 2 (2022)

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Hilbert schemes and $y$–ification of Khovanov–Rozansky homology

Eugene Gorsky and Matthew Hogancamp

Geometry & Topology 26 (2022) 587–678

We define a deformation of the triply graded Khovanov–Rozansky homology of a link L depending on a choice of parameters yc for each component of L, which satisfies link-splitting properties similar to the Batson–Seed invariant. Keeping the yc as formal variables yields a link homology valued in triply graded modules over [xc,yc]cπ0(L). We conjecture that this invariant restores the missing Q TQ1 symmetry of the triply graded Khovanov–Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman’s description of the isospectral Hilbert scheme.

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Khovanov–Rozansky homology, Soergel bimodules, Hilbert schemes
Mathematical Subject Classification 2010
Primary: 14C05, 57M27
Received: 7 January 2020
Revised: 12 February 2021
Accepted: 10 April 2021
Published: 15 June 2022
Proposed: Ciprian Manolescu
Seconded: Paul Seidel, Richard P Thomas
Eugene Gorsky
Department of Mathematics
University of California, Davis
Davis, CA
United States
Matthew Hogancamp
Department of Mathematics
Northeastern University
Boston, MA
United States