ScalingStacks

The paper [25] contains a description of the skein lasagna module for 2-handlebodies (four-manifolds $W$ made of a 0-handle and some 2-handles), where the link $L\subset\partial W$ is empty, or at least local (contained in a $3$-ball). The description is in terms of the Khovanov-Rozansky homology of cables of the attaching link $K$.

In this subsection we extend that description to the case where we attach 2-handles to any four-manifold $W$, to obtain a new manifold $W^{\prime}$. Moreover, we do not impose any restriction on the link $L\subset\partial W$. The formula is very similar to that in [25]. The role of the Khovanov-Rozansky homology $\operatorname{KhR}_{N}$ will be played by the skein lasagna module $\mathcal{S}_{0}^{N}(W;-)$, which can be thought of as a link homology for links in the boundary of $W$. (When $W=B^{4}$, we have $\mathcal{S}_{0}^{N}(W;L)=\operatorname{KhR}_{N}(L)$.)

Let $K_{1},\dots,K_{n}$ be the components of the framed link $K\subset\partial W$ along which the 2-handles are attached. The framing gives diffeomorphisms $f_{i}$ between tubular neighborhoods $\nu(K_{i})$ of each $K_{i}$ and $S^{1}\times D^{2}$. Given $n$-tuples of nonnegative integers

we let $K(k^{-},k^{+})$ denote the framed, oriented cable of $K$ consisting of $k_{i}^{-}$ negatively oriented parallel strands to $K_{i}$ and $k_{i}^{+}$ positively oriented parallel strands. Here, the notion of parallelism for the strands is determined by the framing, that is,

for fixed points $x_{1}^{-},\dots,x_{k_{i}^{-}}^{-},x_{1}^{+},\dots,x_{k_{i}^{+}}^{+}\in D^{2}.$

After attaching 2-handles to $W$ along $K$, we obtain the manifold $W^{\prime}$. Suppose we are given a framed link $L\subset\partial W^{\prime}$. Generically, we can assume that $L$ stays away from the attaching regions of the 2-handles, and therefore we can represent it as a link in $\partial W$, disjoint from (but possibly linked with) $K$. (There are various ways of isotoping $L$ off of the attaching regions; the results of the calculation will be isomorphic.) We let

be the union of $K(k^{-},k^{+})$ and $L$, where we do the cabling on the components of $K$ by choosing the tubular neighborhoods of $K_{i}$ to be disjoint from $L$. (Note that $K(k^{-},k^{+})\cup L$ is not a split disjoint union.)

We seek to express the skein lasagna module $\mathcal{S}_{0}^{N}(W^{\prime};L)$ in terms of $\mathcal{S}_{0}^{N}(W;K(k^{-},k^{+})\cup L)$. To do this, we need to introduce a few more notions.

For each $i$, let $B_{k_{i}^{-},k_{i}^{+}}$ be the subgroup of the braid group on $k_{i}^{-}+k_{i}^{+}$ strands that consists of self-diffeomorphisms of $D^{2}$ rel boundary (modulo isotopy rel boundary) taking the set $\{x_{1}^{-},\dots,x_{k_{i}^{-}}^{-}\}$ to itself and the set $\{x_{1}^{+},\dots,x_{k_{i}^{+}}^{+}\}$ to itself. By taking the product with the identity on $S^{1}$, a braid element $b\in B_{k_{i}^{-},k_{i}^{+}}$ induces a self-diffeomorphism of $D^{2}\times S^{1}$, which can be pulled back (via $f_{i}$) to a self-diffeomorphism of $\nu(K_{i})$. This gives a group action

Let $e_{i}\in{\mathbb{Z}}^{n}$ denote the $i^{th}$ basis vector. Two strands parallel to $K_{i}$, if they have opposite orientations, co-bound a ribbon band $R_{i}$ in $S^{3}$. By pushing $R_{i}$ into $S^{3}\times[0,1]$ so that it is properly embedded there, and taking the disjoint union with the identity cobordisms on the other strands, we obtain an oriented cobordism (still denoted $R_{i}$) from $K(k^{-},k^{+})\cup L$ to $K(k^{-}+e_{i},k^{+}+e_{i})\cup L$. For $d=0,1,\dots,N-1$, we can decorate $R_{i}$ with $d$ dots, and obtain a cobordism map

which changes the bigrading by $(0,2d)$.

Next, recall that we have a decomposition (2) for the skein lasagna module $\mathcal{S}_{0}^{N}(W^{\prime};L)$, according to homology classes in $H_{2}^{L}(W^{\prime};{\mathbb{Z}})$. Let us see how these homology classes are related to the similar ones in $W$. Consider the tubular neighborhood $\nu(K)=\cup_{i}\nu(K_{i})$, which is a union of solid tori. Express $W^{\prime}$ as the union

where $Z$ is the union of the new 2-handles, and $C\cong\nu(K)\times[0,1]$ is a connecting cylinder between $W$ and $Z$. Let also

We identify $\nu(K)$ with $\nu(K)\times\{0\}$ and denote $\nu(K)\times\{1\}$ by $\partial_{-}Z$ (part of the boundary $\partial Z$).

The Mayer-Vietoris sequence for $W^{\prime}$ relative to the union of $W$ and $Z\cup L$ reads

Observe that, by excision, $H_{3}(W^{\prime},W\cup(Z\cup L);{\mathbb{Z}})\cong H_{3}(C,C^{\prime};{\mathbb{Z}})=0$. From here we obtain an exact sequence

Thus, an element in $H_{2}^{L}(W^{\prime};{\mathbb{Z}})\subseteq H_{2}(W^{\prime},L;{\mathbb{Z}})$ can be identified with its image in $H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})$, which we write as a pair $(\alpha,\eta)$.

Let us further identify $H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})$ with ${\mathbb{Z}}^{n}$ by letting the $i$th handle correspond to the coordinate vector $e_{i}$. Then, we write

and let $\alpha^{+}$ denote its positive part and $\alpha^{-}$ its negative part; i.e., $\alpha^{+}_{i}=\operatorname{max}(\alpha_{i},0)$ and $\alpha^{-}_{i}=\operatorname{min}(\alpha_{i},0)$. We also let $|\alpha|=\sum_{i}|\alpha_{i}|$.

Let $r\in{\mathbb{N}}^{n}$ and consider the cable $K(r-\alpha^{-},r+\alpha^{+})$. The fact that $(\alpha,\eta)\in{\mathbb{Z}}^{n}\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})$ is in the kernel of the map to $H_{2}(C,C^{\prime};{\mathbb{Z}})\cong{\mathbb{Z}}^{n}$ in (8) implies the existence of a (unique) class

which is sent to $\eta$ by the natural map to $H_{2}(W,L\cup\nu(K);{\mathbb{Z}}).$

From now on, using the deformation retraction from $\nu(K)$ to $K$, let us think of $\eta$ as a class in $H_{2}(W,K\cup L;{\mathbb{Z}})$.

We will primarily be using the results from this subsection in the case where the role of $W$ is played by

a manifold obtained from a 0-handle by attaching some 1-handles. We denote $W^{\prime}$ by $W_{2}$. Then, $H_{2}(W_{1};{\mathbb{Z}})=0$, so $H_{2}^{L}(W_{1};{\mathbb{Z}})=0$ for any null-homologous $L$, and the decomposition (2) for skein lasagna modules of links in $W_{1}$ is trivial (consists of a single summand). Moreover, in this case an element $(\alpha,\eta)\in H_{2}^{L}(W_{2};{\mathbb{Z}})\subseteq H_{2}(W_{2},L;{\mathbb{Z}})$ is uniquely determined by its image $\alpha$ in $H_{2}(W_{2},W_{1};{\mathbb{Z}})\cong{\mathbb{Z}}^{n}$. Indeed, the exact sequence

show that the component $\eta$ is determined by its image in

The part in $H_{1}(L;{\mathbb{Z}})$ has to be the fundamental class $[L]$, while the part in $H_{1}(\nu(K);{\mathbb{Z}})\cong{\mathbb{Z}}^{n}$ is the image of $\alpha$ under the isomorphisms

Therefore, in this case the class $\eta$ is redundant (being determined by $\alpha$), so we simply drop it from the notation, writing for example $\alpha$ instead of $(\alpha,\eta)$ for the classes in $H_{2}^{L}(W_{2};{\mathbb{Z}})$. With this in mind, the isomorphism from Theorem 3.2 is written as