ScalingStacks

Khovanov [Kh] 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 [La1, La2]. It coincides with Gomi’s trace [Go] for Weyl groups (Webster and Williamson [WeWi]) as well as for dihedral groups [La1].

In the first section, we recall the construction of $2$-braid groups [Rou1], based on complexes of Soergel bimodules. The second section is devoted to Markov traces, and a category-valued version, $2$-Markov traces. We provide a construction using Hochschild cohomology. The third section is devoted to the proof of the Markov property for Hochschild cohomology.