ScalingStacks

Theorem 1.3. Let $W_{1}=\natural^{m}(S^{1}\times B^{3})$ with a nullhomologous link $L\subset\partial W_{1}$ in the boundary that intersects the belt spheres of the 1-handles transversely in $2p_{i}$ points for $1\leq i\leq m$. Let $R\subset S^{3}\setminus\bigsqcup_{i}(B_{i}\cup\overline{B_{i}})$ denote the tangle obtained from $L$ by cutting open along the belt spheres. Then, the skein lasagna module $\mathcal{S}_{0}^{N}(W_{1};L,\mathbbm{k})$ is isomorphic to the quotient

$$\bigoplus_{\begin{subarray}{c}\mathrm{tangles}~T_{i}\\ |\partial T_{i}|=2p_{i}\end{subarray}}\operatorname{KhR}_{N}(R\cup\bigsqcup_{i}(T_{i}\sqcup\overline{T_{i}}),\mathbbm{k})\{(\textstyle\sum_{i}p_{i})(N-1)\}\big/\sim$$

where $\{\cdot\}$ denotes a grading shift, and the relation $\sim$ is given by taking coinvariants for the actions of certain categories $\mathcal{S}_{0}^{N}(B^{3};P_{p_{i}})$ associated to the configurations $P_{p_{i}}$ of $p_{i}$ positively oriented and $p_{i}$ negatively oriented points in $S^{2}=\partial B^{3}$. (See Theorem 4.7 for a more precise statement.)