#### 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
##### Abstract

We define a deformation of the triply graded Khovanov–Rozansky homology of a link $L$ depending on a choice of parameters ${y}_{c}$ for each component of $L$, which satisfies link-splitting properties similar to the Batson–Seed invariant. Keeping the ${y}_{c}$ as formal variables yields a link homology valued in triply graded modules over $ℚ{\left[{x}_{c},{y}_{c}\right]}_{c\in {\pi }_{0}\left(L\right)}$. We conjecture that this invariant restores the missing $Q↔T{Q}^{-1}$ 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.

##### Keywords
Khovanov–Rozansky homology, Soergel bimodules, Hilbert schemes
##### Mathematical Subject Classification 2010
Primary: 14C05, 57M27
##### Publication
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
##### Authors
 Eugene Gorsky Department of Mathematics University of California, Davis Davis, CA United States Matthew Hogancamp Department of Mathematics Northeastern University Boston, MA United States