A geometric construction of colored HOMFLYPT homology

TL;DR

This paper provides a geometric framework for understanding HOMFLYPT homology and its colored variant, connecting algebraic invariants with sheaf cohomology on algebraic groups, and confirming their invariance and categorification properties.

Contribution

It offers a geometric construction of HOMFLYPT homology and its colored version, aligning with existing categorifications and providing new insights into their structure.

Findings

Geometric description of HOMFLYPT homology via sheaf cohomology

Extension of the construction to colored HOMFLYPT homology

Verification that the invariant matches existing categorifications

Abstract

The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All the differentials and gradings which appear in the construction of HOMFLYPT homology are given a geometric interpretation. In fact, with only minor modifications, we can extend this construction to give a categorification of the colored HOMFLYPT polynomial, colored HOMFLYPT homology. We show that it is in fact a knot invariant categorifying the colored HOMFLYPT polynomial and that this coincides with the categorification proposed by Mackaay, Stosic and Vaz.