Anna Beliakova, Krzysztof Putyra, Stephan Wehrli
Abstract: Motivated by topology, we develop a general theory of traces and shadows in
an endobicategory, which is a bicategory $\mathbf C$ with an endobifunctor
$\Sigma\colon \mathbf C \to\mathbf C$. Applying this framework to the
bicategory of Chen-Khovanov bimodules with identity as $\Sigma$ we reproduce
Asaeda-Przytycki-Sikora (APS) homology for links in a thickened annulus. If
$\Sigma$ is a reflection, we obtain the APS homology for links in a thickened
M\"obius band. Both constructions can be deformed by replacing $\Sigma$ with an
endofunctor $\Sigma_q$ such that $\Sigma_q \alpha:=q^{-|\alpha|}\Sigma\alpha$
for any 2-morphism $\alpha$ and identity otherwise, where $q$ is a fixed
invertible scalar. We call the resulting invariant the \emph{quantum link
homology}. We prove in the annular case that this homology carries an action of
$\mathcal U_q(\mathfrak{sl}_2)$, which intertwines the action of cobordisms. In
particular, the quantum annular homology of an $n$-cable admits an action of
the braid group, which commutes with the quantum group action and factors
through the Jones skein relation. This produces a nontrivial invariant for
surfaces knotted in four dimensions. Moreover, a direct computation for torus
links shows that the rank of quantum annular homology groups does depend on the
quantum parameter $q$. Hence, our quantum link homology has a richer structure.
ArXiv: 1605.03523