ScalingStacks

The construction of the 4-manifold invariant $\mathcal{S}^{N}_{0}$ is most straightforward when using the setting of a disklike 4-category or the related notion of a lasagna algebra, a 4-dimensional analog of a planar algebra which we introduce in §5. Indeed, the bigraded abelian group $\mathcal{S}^{N}_{0}(W;L)$ is constructed as a skein module (inspired by the 3-dimensional analogs of Conway, Przytycki [Prz91] and Turaev [Tur91]) spanned by certain decorated surfaces in $W$ bounding $L$, which we call lasagna fillings, modulo skein relations imposed by the operad structure of the lasagna algebra.

More generally, there are bigraded abelian groups $\mathcal{S}^{N}_{i}(W;L)$ for $i\geq 1$ that arise as homology groups of the blob complex defined in [MW12] and can be thought of as higher derived analogs of $\mathcal{S}^{N}_{0}(W;L)$. In fact, the construction of the blob complex was motivated by the idea of using the $\mathcal{S}^{N}_{i}(W;L)$ as tools for computing $\mathcal{S}^{N}_{0}(W;L)$. We can think of $\mathcal{S}^{N}_{i}(W;L)$ as analogous to the $i$-th Hochschild homology $\mathrm{HH}_{i}$, where the input algebra of the Hochschild construction has been replaced by the 4-category derived from $\mathrm{KhR}_{N}$ and the implicit circle in the Hochschild construction has been replaced by the 4-manifold $W$. In particular, when $W$ is the standard 4-ball, so $L$ is a link in the 3-sphere, then $\mathcal{S}^{N}_{0}(B^{4};L)$ is isomorphic to the usual Khovanov–Rozansky homology of $L$, and for $i>0$ the abelian groups are zero.

The Khovanov–Rozansky 4-categories can be fed into the general machinery of [Wal, MW12] to produce fully extended $4{+}\epsilon$-dimensional TQFTs. One consequence of this is that the invariants $\mathcal{S}^{N}_{*}$ satisfy a gluing formula [MW12, Theorem 7.2.1] expressed in terms of a tensor product over a category associated to the gluing locus. In particular, we expect that $\mathcal{S}^{N}_{*}(B^{3}\times S^{1};\{2n\text{ points}\}\times S^{1})$ is related to the Hochschild homology of the $\mathfrak{gl}_{N}$ analog of Khovanov’s arc algebra. For applications of the latter to link homology see Rozansky [Roz10] and Willis [Wil21] $(N=2)$ and Gorsky–Hogancamp–Wedrich [GHW21] ($N=\infty$).

We would like to emphasize that $\mathcal{S}^{N}_{*}$ should be thought of as a categorified analog of the $3{+}\epsilon$-dimensional skein module TQFTs, see Walker [Wal], or the 3-dimensional layers of Crane–Yetter–Kauffman TQFTs [CKY97] at generic $q$, but not the $2{+}1$-dimensional Witten–Reshetikhin–Turaev TQFTs [Wit89, RT91].

Precise relationships along these lines and calculations based on the gluing formula appear in later papers [MN20, MWW2, HRW3].