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

TL;DR

This paper introduces a deformation of Khovanov-Rozansky homology depending on component parameters, conjectures its symmetry restoration, and connects it to Hilbert schemes, with explicit computations for the full twist.

Contribution

It defines a new link homology invariant with parameters, conjectures its symmetry properties, and links it to Hilbert schemes and Haiman's ideals, advancing the understanding of link invariants and geometric connections.

Findings

Invariant matches ideals in Haiman's description of the isospectral Hilbert scheme.

Conjectured to restore missing symmetry in Khovanov-Rozansky homology.

Explicit computations for the full twist demonstrate the invariant's properties.

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 $\mathbb{Q}[x_c,y_c]_{c\in \pi_0(L)}$. We conjecture that this invariant restores the missing $Q\leftrightarrow TQ^{-1}$ 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.