ScalingStacks

Remark 4.17. One can also consider analogs of the skein modules $\mathcal{S}_{0}^{N}$ based on equivariant or deformed versions of $\mathfrak{gl}_{N}$ homology. For example, in one common choice for $N=2$ one works over $R^{\prime}=\mathbbm{k}[X,\alpha]/(X^{2}=\alpha)$. We can also try to compute the bigraded zeroth Hochschild homology of the 3-ball category with two points and of its ambient category $H^{\bullet}(\operatorname{Ch}_{\mathrm{dg}}(R^{\prime}\mathrm{-mod}^{\mathrm{gr.fr.}}))^{*}$ in this setting. We have already listed the indecomposable of the latter above: the chain complexes $C^{k}$. For $k\geq 1$ the enriched isomorphism algebra of the complex $C^{k}$ is isomorphic to $R^{\prime}[\eta]/(X^{k}=0)$ where $\eta$ is of bidegree $(1,2k)$. The trace classes of $\eta$ and its multiples are zero. Moreover, the trace class of $X^{x}$ is zero for every $x>0$. This leaves the trace classes of the identities of $C^{k}$ for $k\geq 0$ and the trace class of $X_{C^{0}}$ as linearly independent — the zeroth Hochschild homology is not locally finite-dimensional. However, it is currently not known which $C^{k}$ appear in complexes of $(1,1)$-tangles. A copy of $C^{3}$ appears in [24].