ScalingStacks

Having built a disklike 4-category we immediately obtain an alternative description of the skein module $\mathcal{S}^{N}_{0}(W;L)$ for a link in the boundary of any oriented smooth 4-manifold $W$, as first introduced in §5.2.

This is the construction from [MW12, §6.3], which describes $\mathcal{S}^{N}_{0}(W;L)$ as a colimit, taken over all ways of decomposing a 4-manifold $W$ into a gluing of closed balls (with some regularity conditions on the ways these balls meet). For any such decomposition, we draw compatible links in the boundaries of each of the balls (i.e. if two balls meet along some 3-manifold, the intersections of the two links with that 3-manifold are tangles, and identical, and a similar condition holds for any ball meeting $\partial W$). Then the bigraded abelian group at such a decomposition is the direct sum, over the choices of link labels, of the tensor products of the Khovanov–Rozansky homologies of each link. The arrows in the colimit diagram are ways of coarsening the decomposition by gluing several balls together into a single ball. The gluing maps for a disklike 4-category provide morphisms of bigraded abelian groups. Finally, the skein module invariant $\mathcal{S}^{N}_{0}(W;L)$ associated to $W$ is just the colimit of this diagram.

We will leave it as an exercise to the interested reader to verify that these two constructions actually give the same result!

Our motivation for introducing the disklike 4-category is that the construction of [MW12, §6.3] actually gives much more. Associated to any link $L$ in the boundary of a 4-manifold $W$, we obtain the blob complex (with coefficients in the disklike 4-category $\mathsf{KhR}_{N}$), which we write as ${\mathcal{B}}_{*}(\mathsf{KhR}_{N})(W;L)$. (One approach to the definition of this complex is by replacing the colimit described above with an appropriate homotopy colimit, see [MW12, §7].) This has a new homological grading, unrelated to the internal homological grading from Khovanov-Rozansky homology. The 0-th homology of this complex recovers the bigraded abelian group $\mathcal{S}^{N}_{0}(W;L)$, but the higher blob homology groups, denoted by $\mathcal{S}^{N}_{i}(W;L)$ for $i>0$, potentially carry further information.

Attempting any calculations of this invariant, or of its 0-th homology in either formulation, remains beyond the scope of this paper, and developing appropriate computational tools is an open problem for future work (e.g. [MN20]). One such tool should come from a categorification of the $\mathfrak{gl}_{N}$ skein relation, namely the skein exact triangle for Khovanov–Rozansky chain complexes in ${\mathbb{R}}^{3}$, which induces a long exact sequence on homology groups. For a skein triple of links in the boundary of some interesting 4-manifold $W$ we have every reason to expect that the corresponding sequence on the level of the skein module $\mathcal{S}^{N}_{0}$ is no longer exact. We do, however, obtain long exact sequences on the level of the blob complex, which give rise to a spectral sequence that relates the skein modules $\mathcal{S}^{N}_{0}$ for the three links. In fact, the study of these spectral sequences was the original motivation for the blob complex (however ahistorical this might seem, given the publication dates).