Annular Evaluation and Link Homology

Abstract

We use categorical annular evaluation to give a uniform construction of both ${\mathfrak{sl}}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield ${\mathfrak{gl}}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalgebra ${\mathfrak{gl}}_{0|n}$, both for links in $S^3$ and in the thickened annulus. In the $n=2$ case, this produces a categorification of the Jones polynomial that we show is distinct from Khovanov homology, and gives a finite-dimensional categorification of the colored Jones polynomial. This behavior persists for general $n$. Our approach yields simple constructions of spectral sequences relating these theories, and emphasizes the roles of super vector spaces, categorical traces, and current algebras in link homology.

47 pages, many figures