Links in genus-g handlebodies are assigned a triply-graded homology built from singular Soergel bimodules and Hochschild cohomology, generalizing colored HOMFLYPT homology.
Khovanov-Rozansky homology and 2-braid groups
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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.
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HOMFLYPT homology for links in handlebodies via type A Soergel bimodules
Links in genus-g handlebodies are assigned a triply-graded homology built from singular Soergel bimodules and Hochschild cohomology, generalizing colored HOMFLYPT homology.