ScalingStacks

We start by reviewing the construction of skein lasagna modules from [27, Section 5.2].

Following [27] and [25], for a framed link $L\subset{\mathbb{R}}^{3}$, we write

for the $\mathfrak{gl}_{N}$ version of Khovanov-Rozansky homology. Here, $i$ denotes the homological grading and $j$ denotes the quantum grading.

If we have an oriented manifold $S$ diffeomorphic to the standard $3$-sphere $S^{3}$, and a framed link $L\subset S$, we can define a canonical invariant $\operatorname{KhR}_{N}(S,L)$ as in [27, Definition 4.12]. We sometimes drop $S$ from the notation and simply write $\operatorname{KhR}_{N}(L)$.

Given a framed cobordism $\Sigma\subset S^{3}\times[0,1]$ from $L_{0}$ to $L_{1}$, there is an induced map

which is homogeneous of bidegree $(0,(1-N)\chi(\Sigma))$.

Let $W$ be a four-manifold and $L\subset\partial W$ a framed link. A lasagna filling $F=(\Sigma,\{(B_{i},L_{i},v_{i})\})$ of $W$ with boundary $L$ consists of

• •

  A finite collection of disjoint $4$-balls $B_{i}$ (called input balls) embedded in the interior or $W$;

• •

  A framed oriented surface $\Sigma$ properly embedded in $W\setminus\cup_{i}B_{i}$, meeting $\partial W$ in $L$ and meeting each $\partial B_{i}$ in a link $L_{i}$; and

• •

  for each $i$, a homogeneous label $v_{i}\in\operatorname{KhR}_{N}(\partial B_{i},L_{i}).$

The bidegree of a lasagna filling $F$ is

If $W$ is a $4$-ball, we can define a cobordism map

and an evaluation

We define the skein lasagna module as the bigraded abelian group

where $\sim$ is the transitive and linear closure of the following relation:

1. (a)

   Linear combinations of lasagna fillings are set to be multilinear in the labels $v_{i}$;

2. (b)

   Furthermore, two lasagna fillings $F_{1}$ and $F_{2}$ are set to be equivalent if $F_{1}$ has an input ball $B_{i}$ with label $v_{i}$, and $F_{2}$ is obtained from $F_{1}$ by replacing $B_{i}$ with another lasagna filling $F_{3}$ of a $4$-ball such that $v_{i}=\operatorname{KhR}_{N}(F_{3})$, followed by an isotopy rel $\partial W$ (where the isotopy is allowed to move the input balls):

For future reference, here is a useful lemma.

Skein lasagna modules decompose according to relative homology classes, as noted in [25, Section 2.3]:

Observe that in the case where $L$ is not null-homologous in $W$ (i.e. $[L]\neq 0\in H_{1}(W;{\mathbb{Z}})$), then there are no lasagna fillings, so $\mathcal{S}_{0}^{N}(W;L)=0$. When $[L]=0\in H_{1}(W;{\mathbb{Z}})$, consider the boundary map in the long exact sequence of the pair $(W,L)$:

The only classes $\alpha\in H_{2}(W,L;{\mathbb{Z}})$ that can contribute non-trivially are those that map to the fundamental class $[L]\in H_{1}(L;{\mathbb{Z}})$ under $\partial$. Let us introduce the notation

Note that, using the long exact sequence of the pair, the difference of two classes in $H_{2}^{L}(W;{\mathbb{Z}})$ can be identified with an element of $H_{2}(W;{\mathbb{Z}})$. Thus, $H_{2}^{L}(W;{\mathbb{Z}})$ is a torsor over $H_{2}(W;{\mathbb{Z}})$; it can be identified with the latter group after choosing a base element in $H_{2}^{L}(W;{\mathbb{Z}})$.

The decomposition  (1) becomes

We will use the decomposition (2) in the case of a general link $L$; when $[L]\neq 0$, we have $H_{2}^{L}(W;{\mathbb{Z}})=\emptyset$ and $\mathcal{S}_{0}^{N}(W;L)=0$.