Khovanov-Rozansky homology and 2-braid groups

TL;DR

This paper proves that Khovanov's construction of link invariants via Hochschild cohomology of 2-braid groups is valid and extends to finite Coxeter groups, providing a categorical framework for these invariants.

Contribution

It offers a direct proof of Khovanov's link invariants construction and generalizes it to finite Coxeter groups with a categorical perspective.

Findings

Khovanov's construction yields valid link invariants.

The invariants form a Markov '2-trace' in derived categories.

Extension of the invariants to finite Coxeter groups.

Abstract

Khovanov has given a construction of the Khovanov-Rozansky link invariants (categorifying the HOMFLYPT invariant) using Hochschild cohomology of 2-braid groups. We give a direct proof that his construction does give link invariants. We show more generally that, for any finite Coxeter group, his construction provides a Markov "2-trace", and we actually show that the invariant takes value in suitable derived categories. This makes more precise a result of Trafim Lasy who has shown that, after taking the class in K_0, this provides a Markov trace.