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Homological invariants such as Khovanov homology [17] and Khovanov-Rozansky homology [21] are at the center of modern knot theory. These invariants were originally defined for links in ${\mathbb{R}}^{3}$. Extending them to links in arbitrary $3$-manifolds is a problem that garnered much attention recently, from various perspectives (categorification at roots of unity [20, 9, 28], theoretical physics [34, 14, 13], etc.)

One such extension was introduced in [27], based on higher category theory and using the concept of blob homology [26]. Given a smooth, oriented, compact four-manifold $W$ and a framed oriented link $L$ in the boundary $\partial W$, the construction in [27] associates to the pair $(W,L)$ a homology theory graded by ${\mathbb{Z}}^{3}\times H_{2}(W,L;{\mathbb{Z}})$ and denoted $\mathcal{S}^{N}_{*}(W;L)$. One of the three integer gradings is called the blob degree, and for our purposes we will focus on the theory in blob degree zero, $\mathcal{S}_{0}^{N}(W,L)$. This is called the skein lasagna module of $(W,L)$ and has a relatively simple definition, reminiscent of the definition of the skein module of a $3$-manifold. The skein lasagna module is defined as the span of the lasagna fillings of $W$ with boundary $L$, modulo an equivalence relation. The lasagna fillings are certain decorated surfaces connecting $L$ to other links in the boundaries of $4$-balls inside $W$, and the equivalences come from cobordism maps in Khovanov-Rozansky homology.

Skein lasagna modules are challenging to compute. It was proved in [27] that when $W=B^{4}$, the invariant $\mathcal{S}_{0}^{N}(B^{4};L)$ coincides with the Khovanov-Rozansky homology of the link $L$. Further computational methods were developed in [25], with a focus on 2-handlebodies (four-manifolds obtained from $B^{4}$ by attaching 2-handles). This allowed the calculation of the skein lasagna modules (in some gradings) for four-manifolds such as the complex projective plane, and disk bundles over $S^{2}$.

In this paper, building on the work in [27] and [25], we give a new formula for the skein lasagna module of a link in the boundary of an arbitrary four-manifold. We start by choosing a handle decomposition for the four-manifold. For simplicity, we may assume that we have a single 0-handle. We then study how the skein lasagna module changes under adding handles. Disjoint unions, 4-handles and many cases of 2-handles were already studied in [25], so the main thing left is to understand 1- and 3-handles.

With regard to 3-handles, we have the following:

Next, we combine Theorem 1.1 with the treatment of 2-handles in [25] to get a general result, reducing the calculation of the skein lasagna module to the case of 1-handles.

Recall that in [25], the skein lasagna module of a 2-handlebody was shown to be isomorphic to the so-called cabled Khovanov-Rozansky of the attaching link for the 2-handles; this is obtained from the Khovanov-Rozansky homologies of the cables of this attaching link, modulo certain cobordism relations. We define an analogue of the cabled Khovanov-Rozansky homology for two links $K,L$ in the boundary of $W_{1}=\natural^{m}(S^{1}\times B^{3})$ (and, more generally, any other four-manifold); we call this the cabled skein lasagna module $\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N}(W_{1};K,L)$.

The cabled skein lasagna module $\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N}(W_{1};K,L)$ is constructed from the invariants $\mathcal{S}_{0}^{N}(W_{1};K(a,b)\cup L)$ where $K(a,b)\cup L$ is a family of framed links in $\partial W_{1}=\#^{m}(S^{1}\times S^{2})$ consisting of $L$ and cables $K(a,b)$ of the attaching link $K$ for the 2-handles. Thus, Theorem 1.2 allows us to express $\mathcal{S}_{0}^{N}(W_{4};L)$ in terms of skein lasagna modules of links in $\partial W_{1}$ (and maps between them).

The second half of our paper studies in more detail the skein lasagna modules for links in $\partial W_{1}$ where $W_{1}=\natural^{m}(S^{1}\times B^{3})$. We work with coefficients in a field $\mathbbm{k}$. By cutting along the cocores of the 1-handles, we reduce the problem of computing $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ to a problem about skein lasagna modules for the (boundary of the) 0-handle $B^{4}$ with a family of framed links related to $L$. For links in $B^{4}$, the invariant $\mathcal{S}_{0}^{N}$ is simply the Khovanov-Rozansky homology $\operatorname{KhR}_{N}$.

Furthermore, we will show that the isomorphisms from Theorem 1.3 are functorial in the following sense: They allow an expression of maps associated to cobordisms $S\subset\partial W_{1}\times I$ between links $S\colon L\to L^{\prime}$ in $\partial W_{1}=\#^{m}(S^{1}\times S^{2})$ in terms of components computed entirely from maps associated to link cobordisms in $S^{3}$.

By combining Theorems 1.2 and 1.3 (plus the functoriality statement), we thus obtain a recipe for expressing the lasagna skein modules of any four-manifold in terms of Khovanov–Rozansky homologies of links in $S^{3}$ and maps associated to cobordisms in $S^{3}\times I$. The invariant is a quotient of a (typically infinite) direct sum of homologies of links by a subspace defined in terms of link cobordism maps.

Specializing the setting of Theorem 1.3 to the case of a single 1-handle, we consider the link $S^{1}\times P_{p}\subset S^{1}\times B^{3}$ consisting of $2p$ parallel circles, with $p$ of them oriented one way and $p$ the other way. We prove that $\mathcal{S}_{0}^{N}(S^{1}\times B^{3},S^{1}\times P_{p})$ is isomorphic to the zeroth Hochschild homology of the category $\mathcal{S}_{0}^{N}(B^{3};P_{p_{i}})$. From here we get the following explicit calculation for $N=2$.

Using methods analogous to those employed in part (a), we also show that $\mathcal{S}_{0}^{2}(S^{1}\times S^{3},\mathbbm{k})$ is one-dimensional; see Corollary 4.2. For part (c), we actually show that $\mathcal{S}_{0}^{2}(S^{1}\times B^{3},S^{1}\times P_{p},\mathbbm{k})$ is infinite dimensional in bidegree $(0,0)$. This answers in the negative Question 1.7 from [25], about whether skein lasagna modules are always locally finite dimensional, i.e., finite dimensional in each fixed bidegree and homology class.

This still leaves open the following:

For $W_{1}=\natural^{m}(S^{1}\times B^{3})$, one can view the skein lasagna module $\mathcal{S}_{0}^{2}(W_{1};L)$ as a variant of Khovanov homology for links $L$ in $\#^{m}(S^{1}\times S^{2})$. Another version of Khovanov homology for these links was constructed by Rozansky (for $m=1$) in [31], and Willis [33] for arbitrary $m$. The Rozansky–Willis homology $H^{*,*}_{\operatorname{RW}}(L)$ is finitely generated in each bidegree and, thus, different from our theory. We expect that $H^{*,*}_{\operatorname{RW}}(L)$ appears on the $E_{2}$ page of a spectral sequence converging to $\mathcal{S}_{0}^{2}(W_{1};L)$. See Section 4.6 for a further discussion and Section 4.7 for a conjectural extension of the Rozansky–Willis homology to links in the boundary of other four-manifolds.

Organization of the paper. In Section 2 we go over a few preliminaries about skein lasagna modules and Kirby diagrams. In Section 3 we study the behavior of skein lasagna modules under attaching 2- and 3-handles, proving Theorems 1.1 and  1.2. In Section 4 we focus on 1-handles, and prove Theorems 1.3 and  1.5.

Conventions. All the manifolds considered in this paper will be smooth, compact, and oriented. All links and surfaces are oriented and normally framed.

Acknowledgements. This paper builds on previous joint work and many enlightening conversations of KW and PW with Scott Morrison, without which this paper probably would not exist. We would also like to thank Matthew Hogancamp and Ikshu Neithalath for helpful comments on a draft of this paper.