ScalingStacks

Proof. An element $v\in\mathcal{S}_{0}^{N}(W;K(r-\alpha^{-},r+\alpha^{+})\cup L,\eta^{r})$ is represented by a linear combination of lasagna fillings $(\Sigma,\{(B_{i},L_{i},v_{i}\})$ in $W$, where $\partial\Sigma=K(r-\alpha^{-},r+\alpha^{+})\cup L\cup(\cup_{i}L_{i})$. We define $\Phi(v)$ to be the class of the linear combination of lasagna fillings with the same input data $\{(B_{i},L_{i},v_{i}\}$ as $v$, but with the surfaces given by attaching to each $\Sigma$ (along its boundary) the disjoint union of $r_{i}-\alpha_{i}^{-}$ negatively oriented discs parallel to the core of $i^{th}$ 2-handle and $r_{i}+\alpha_{i}^{+}$ positively oriented such discs (union over all $i$).

We also define a map $\Phi^{-1}$ in the opposite direction, as follows. Let $F$ be a lasagna filling in $W^{\prime}$ with surface $\Sigma$. We isotope the input balls of $F$ to be inside $W$, and isotope the surface $\Sigma$ such that its intersection with the 2-handles consists of several disks parallel to their cores. Removing these disks produces a lasagna filling of $W$ with boundary on a link of the form $K(r-\alpha^{-},r+\alpha^{+})\cup L$. We let this be $\Phi^{-1}(F)$.

The proofs that $\Phi$ and $\Phi^{-1}$ are well-defined and inverse to each other are similar to the proof of Theorem 1.1 in [25], which dealt with the case $W=B^{4}$ and $L=\emptyset$. The extension to arbitrary $W$ and $L$ is obtained by replacing the Khovanov-Rozansky homologies $\operatorname{KhR}_{N}$ with the skein lasagna modules in $W$. (In the formulation here, the proof of the statement is even slightly clearer since it relates lasagna skein modules with lasagna skein modules. In particular, we do not have to choose standard lasagna fillings with “slighly smaller input balls”, as these were only required when comparing $\mathcal{S}_{0}^{N}(B^{4},-)$ with $\operatorname{KhR}_{N}$.) ∎