Link splitting deformation of colored Khovanov--Rozansky homology

TL;DR

This paper introduces a multi-parameter deformation of colored Khovanov--Rozansky homology, extending its framework to braids and establishing new splitting properties and connections to Hilbert schemes.

Contribution

It develops a novel multi-parameter deformation of colored Khovanov--Rozansky homology and extends the theory to braids using curved complexes of singular Soergel bimodules.

Findings

Computed invariants of colored Hopf links using ideals from Haiman determinants.

Established link splitting properties for the deformed homology.

Formulated conjectures relating homology to Hilbert schemes.

Abstract

We introduce a multi-parameter deformation of the triply-graded Khovanov--Rozansky homology of links colored by one-column Young diagrams, generalizing the "$y$-ified" link homology of Gorsky--Hogancamp and work of Cautis--Lauda--Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2-category of curved complexes of type $A$ singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from arXiv:2107.08117 and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply-graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov--Rozansky homology and Hilbert schemes.