Hilbert schemes and y–ification of Khovanov–Rozansky homology

Gorsky, Eugene; Hogancamp, Matthew

doi:10.2140/gt.2022.26.587

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

Eugene Gorsky and Matthew Hogancamp

Geometry & Topology 26 (2022) 587–678

DOI: 10.2140/gt.2022.26.587

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 $ℚ {[x_{c}, y_{c}]}_{c \in π_{0} (L)}$ . We conjecture that this invariant restores the missing $Q \leftrightarrow T Q^{- 1}$ 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

References

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

Volume 26, issue 2 (2022)