ScalingStacks

Definition 3.1. The cabled skein lasagna module of $K\subset\partial W$ at level $\alpha$ and in class $\eta$ is

$$\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W;K,L,\eta)=\Bigl(\bigoplus\limits_{r\in{\mathbb{N}}^{n}}\mathcal{S}_{0}^{N}(W;K(r-\alpha^{-},r+\alpha^{+})\cup L,\eta^{r})\{(1-N)(2|r|+|\alpha|)\}\Bigr)/\sim$$

where the equivalence $\sim$ is the transitive and linear closure of the relations

(9)

$$\beta_{i}(b)v\sim v,\ \ \psi^{[d]}_{i}(v)\sim 0\text{ for }d<N-1,\ \ \psi^{[N-1]}_{i}(v)\sim v$$

for all $i=1,\dots,n$; $b\in B_{k_{i}^{-},k_{i}^{+}}$, and $v\in\mathcal{S}_{0}^{N}(W;K(r-\alpha^{-},r+\alpha^{+})\cup L,\eta^{r}).$