ScalingStacks

In [31], Rozansky defined a Khovanov-type homology theory for (null-homologous) links in $S^{1}\times S^{2}$. His construction was generalized by Willis in [33] to null-homologous links in $Y=\#^{m}(S^{1}\times S^{2})$ for any $m$. We will denote the Rozansky-Willis homology of $L\subset Y$ by $H^{*,*}_{\operatorname{RW}}(L)$. Just like the skein lasagna module $\mathcal{S}_{0}^{2}(W_{1},L)$, the invariant $H^{*,*}_{\operatorname{RW}}(L)$ can be computed from a Kirby diagram for $W_{1}=\natural^{m}(S^{1}\times B^{3})$ including the link $L$, so it is a natural question whether they are related.

The first observation is that the two invariants are not always isomorphic. Indeed, in any specific bidegree, $H^{*,*}_{\operatorname{RW}}(L)$ is defined as the Khovanov homology of the link in $S^{3}$ obtained from $L$ by adding sufficiently many twists in place of the 1-handles. It follows that $H^{*,*}_{\operatorname{RW}}(L)$ has finite rank in each bidegree, whereas this may not hold for $\mathcal{S}_{0}^{2}(W_{1},L)$, as we have seen in Theorem 4.23. Another concrete example is for $m=1$, where $L=S^{1}\times P_{1}$ yields a $4$-dimensional lasagna skein module according to Proposition 4.15, but $H^{*,*}_{\operatorname{RW}}(L)\cong\operatorname{HH}_{\bullet}(\mathbbm{k}[X]/(X^{2}))$ is infinite-dimensional.

However, $H^{*,*}_{\operatorname{RW}}(L)$ and $\mathcal{S}_{0}^{2}(W_{1},L)$ are conceptually similar, as both arise as the Hochschild homology of a chain complex associated to a tangle $T$ that closes to the link $L$:

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  $H^{*,*}_{\operatorname{RW}}(L)$ is computed as the Hochschild homology of a dg bimodule (for a tensor product of $m$ of Khovanov’s arc rings) associated to the tangle $T$, as defined for $m=1$ by Khovanov in [18] and extended by parabolic induction to $m>1$. Here the homological degree of the dg bimodule gets mixed with the Hochschild degree, and so the resulting invariant is a bigraded vector space.

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  $\mathcal{S}_{0}^{2}(W_{1},L)$ can be computed via Theorem 4.7 (and for $m=1$ even more concretely in Corollary 4.10) as the zeroth Hochschild homology of an equivalent dg bimodule; see Remark 4.13 for the comparison. In fact, the higher blob homology from [27], which does not play a role for skein lasagna modules, corresponds to higher Hochschild homology. The main difference, however, is that the dg bimodule is not considered as an object of a dg or triangulated category, but of the linear cohomology category. Accordingly, the full blob homology is triply-graded, with the blob/Hochschild grading separated from the homological grading.

Based on this comparison, one may expect $\mathcal{S}_{0}^{2}(W_{1},L)$ and, more generally, the full blob homology $\mathcal{S}^{N}_{*}(W_{1};L)$ to appear on the $E_{2}$ page of a spectral sequence converging to $H^{*,*}_{\operatorname{RW}}(L)$. Suppose that one can find a suitable projective resolution in terms of tangle complexes, which simultaneously allows the computation of blob homology as well as the dg version of Hochschild homology. Then, by tensoring with the dg bimodule associated to the tangle, one obtains a double complex of (quantum) graded vector spaces, where the vertical differential carries Hochschild degree and the horizontal differential carries homological degree. The homology of the total complex would compute $H^{*,*}_{\operatorname{RW}}(L)$. To obtain $\mathcal{S}_{0}^{2}(W_{1},L)$, one first takes homology in the rows (thus computing the Khovanov homologies of links of the form $T_{i}\cup\overline{T}$ where $T_{i}$ appears in the resolution), and only then the zeroth homology of the induced differential coming from the resolution. We will not pursue this comparison further in the present paper, but remark that there is precedent for interesting invariants appearing on $E_{2}$ pages of spectral sequences that come from separating Hochschild and homological degrees, namely the triply-graded HOMFLYPT link homology; see [29, Section 6].

In general, one does not expect a map from the $E_{2}$ page of a spectral sequence to its $E_{\infty}$ page. However, since $\mathcal{S}_{0}^{2}(W_{1},L)$ appears as the lowest row on the $E_{2}$ page, the above discussion suggests the existence of a natural map

In the following we propose a candidate for such a map.

In Willis’s construction of $H^{*,*}_{\operatorname{RW}}(L)$, we represent $\partial W_{1}=Y$ by $m$ pairs of spheres in the plane, with the spheres in each pair being identified (that is, we add a handle). This is the same as the usual Kirby diagram of $W_{1}$. The link $L$ may intersect each handle a number of times, as in this picture:

Let $L(n_{1},\dots,n_{m})$ be the link in $S^{3}$ obtained from $L$ by inserting $n_{i}$ full twists in place of the $i^{\operatorname{th}}$ handle, as shown here:

The homology $H^{*,*}_{\operatorname{RW}}(L)$ can be computed as the Khovanov homology of the link $L(n_{1},\dots,n_{m})$ for $n_{i}\gg 0$, with some suitable shifts in grading. Note that $L(n_{1},\dots,n_{m})$ depends on the choice of a path between the attaching spheres of each 1-handle; however, it can be shown that $H^{*,*}_{\operatorname{RW}}(L)$ is independent of these choices up to isomorphism.

Consider now the skein lasagna module $\mathcal{S}_{0}^{2}(W_{1},L)$. Let us attach an $n_{i}$-framed 2-handle through the $i^{\operatorname{th}}$ 1-handle:

The 2-handles cancel the corresponding 1-handles, so the result is a Kirby diagram for $B^{4}$, whose boundary is $S^{3}$. The link $L$ becomes $L(n_{1},\dots,n_{m})\subset S^{3}$, as can be seen by doing a series of handle slides of the arcs of $L$ over the 2-handle:

where in the last step we cancelled the handles. (Compare Figure 5.13 in [10].)

The 2-handle attachments give a cobordism $Z$ from $Y=\#^{m}(S^{1}\times S^{2})$ to $S^{3}$. There is also an embedded annular cobordism $S\subset Z$ from $L$ to $L(n_{1},\dots,n_{m})$. As discussed in Section 2.2, these cobordisms induce a map on skein lasagna modules:

Our conjecture is that these maps stabilize as $n_{i}\to\infty$, giving a well-defined morphism from $\mathcal{S}_{0}^{2}(W_{1},L)$ to $H^{*,*}_{\operatorname{RW}}(L)$.