ScalingStacks

Consider the process of cutting a lasagna filling $F$ of $W_{1}=\natural^{m}(S^{1}\times B^{3})$ with boundary $L$ along the cocores $C_{i}\cong\mathrm{pt}\times B^{3}$ of the 1-handles for $1\leq i\leq m$. Let us assume that the lasagna sheet $\Sigma$ of $F$ intersects the cocores transversely in tangles $T_{i}:=\Sigma\cap C_{i}$. In particular, the link $L$ intersects the belt spheres $S_{i}:=\partial C_{i}$ geometrically in $2p_{i}$ points, the boundary points of the tangle $T_{i}$. The algebraic intersection numbers are all zero, since $L$ is null-homologous, as witnessed by $F$. In this way, we obtain a lasagna filling $\mathrm{cut}(F)$ of $W_{1}\setminus\bigsqcup_{i}n(C_{i})\cong B^{4}$ with boundary link

The latter is obtained by cutting $L$ open at the $2p_{i}$-tuples of boundary points and inserting copies of the tangles $T_{i}$ and $\overline{T_{i}}$, schematically:

Of course, the procedure of cutting lasagna fillings does not describe a well-defined map on the level of $\mathcal{S}_{0}^{N}$ since it does not respect the skein relations. Instead we consider the reverse operation.

The process of gluing a lasagna filling works as follows. Let $F^{\prime}$ be a lasagna filling of $B^{4}$ with boundary link $L_{T}$ as above; i.e., inside $S^{3}=\partial B^{4}$ we have $m$ pairs of embedded 3-balls $B_{i}\cup\overline{B_{i}}$, such that $L_{T}\cap B_{i}=T_{i}$ and $L_{T}\cap\overline{B_{i}}=\overline{T}_{i}$ for $1\leq i\leq m$. Denote the numbers of boundary points by $2p_{i}:=|\partial T_{i}|$. Now we attach $m$ 1-handles with core-parallel lasagna sheets $I\times T_{i}\subset I\times B^{3}$ along the $B_{i}\cup\overline{B_{i}}\cong S^{0}\times B^{3}$ to obtain a lasagna filling of $W_{1}$ with boundary $L$. Since the relations in $\mathcal{S}_{0}^{N}$ are local, this induces a map:

The grading shift is there to compensate the change in Euler characteristic of the surfaces in lasagna fillings upon gluing.

It follows that the gluing maps from (17) assemble to a surjective map from a direct sum of shifts of $\mathcal{S}_{0}^{N}(B^{4};L_{T},\mathbbm{k})$ to $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$. Here, the sum is indexed by all ways of writing $L$ as a contraction of links $L_{T}$ obtained by drilling out pairs of tangles $T_{i}\cup\overline{T_{i}}$ and resealing the boundary points across the 1-handles. It remains to describe the kernel.

The statement of Lemma 4.6 can be paraphrased as: the choice of a $3$-ball with point configuration $P_{p}$ in $\partial W$ equips $\mathcal{S}_{0}^{N}(W;-,\mathbbm{k}):=\bigoplus_{L}\mathcal{S}_{0}^{N}(W,L,\mathbbm{k})$ with the structure of a bigraded module for the category $\mathcal{S}_{0}^{N}(B^{3};P_{p})$. (Here the direct sum is taken over all links $L$ that intersect the boundary of the chosen $3$-ball in the fixed configuration $P_{p}$.)

Theorem 4.7 can also be summarized by saying that $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ is computed by the zeroth Hochschild homology of a tensor product of $3$-ball categories, namely one for each handle, with coefficients in a bimodule associated to the tangle $R$ that results from $L$ by cutting open along the belt spheres. We will discuss the details of this perspective in a special case in Section 4.3.

Finally we comment on the functoriality of the 1-handle formula from Theorem 4.7. We have seen that $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ for $W_{1}=\natural^{m}(S^{1}\times B^{3})$ is a colimit of link homologies for links in $S^{3}$, which result from cutting $L$ along belt spheres and inserting pairs of tangles. Now consider a link cobordism $S\subset\partial W_{1}\times I=:Z$ from $L\subset W_{1}$ to $L^{\prime}\subset W_{1}^{\prime}$ where $W_{1}^{\prime}=W_{1}\cup Z$. We claim that the induced map

can also be expressed in terms of cobordism maps between links in $S^{3}$. Recall that the cobordism map $\Psi_{Z;S}$ sends a lasagna filling $F$ of $W_{1}$ to the composite lasagna filling $F\cup S$ of $W_{1}\cup Z$. In a generic situation, cutting the cocores has the following local model. Here we display the filling $F$ in the inner tube and $S$ in the outer, spherical shell.

Let $S_{i}$ denote the tangle in $S^{2}\times I$ that occurs as the intersection of $S$ with the $i$th cocore and $R^{\prime}\subset S^{3}\setminus\bigsqcup_{i}(B_{i}\cup\overline{B_{i}})$ the tangle obtained from $L^{\prime}$ by cutting open along the belt spheres of $W_{1}^{\prime}$. Denote by $2p^{\prime}_{i}=|\partial S_{i}|-2p_{i}$ the number of outer boundary point of $S_{i}$. Then the cobordism $\Sigma$ obtained from $S$ by cutting along the annuli, which are the intersection of $Z$ with the cocores of 1-handles in $W_{1}^{\prime}$, induces a cobordism map:

We claim that these components describe $\Psi_{Z;S}$ in terms of the colimit formulas (left-hand sides) from Theorem 4.7. To see this we first observe that the unequal grading shifts guarantee that the components have the same degree as $\Psi_{Z;S}$ (we have $\chi(\Sigma)=\chi(S)+\sum_{i}(p_{i}+p^{\prime}_{i})$ and $\Sigma$ is glued to $\mathrm{cut}(F)$ along $p_{i}$ interval segments). Next we observe that after composing with the projection-inclusion into the colimit formula for $\mathcal{S}_{0}^{N}(W_{1}^{\prime};L^{\prime},\mathbbm{k})$, the resulting map no longer depends on the chosen location of cocores to cut. Moreover, the subspace factored out in the colimit formula for $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ is annihilated by the map thus defined. Thus the components described above define a map $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})\to\mathcal{S}_{0}^{N}(W_{1}^{\prime};L^{\prime},\mathbbm{k})$, and by construction this agrees with $\Psi_{Z;S}$.