ScalingStacks

Theorem 3.10. Consider four-manifolds $W_{1}\subseteq W_{2}\subseteq W_{3}\subseteq W_{4}$ where

• •

  $W_{1}=\natural^{m}(S^{1}\times B^{3})$ is the union of $m$ 1-handles;

• •

  $W_{2}$ is obtained from $W_{1}$ by attaching $n$ two-handles along a framed link $K$;

• •

  $W_{3}$ is obtained from $W_{2}$ by attaching $p$ three-handles along spheres $S_{1},\dots S_{p}$;

• •

  $W_{4}$ is obtained from $W_{3}$ by attaching some four-handles.

Consider also a framed link $L\subset\partial W_{4}$. We represent $W_{4}$ by a Kirby diagram, viewing $K\cup L$ as a link in $\partial W_{1}$, and the spheres $S_{i}$ in terms of surfaces $\Sigma_{j}$ on $\partial W_{1}$ with $\partial\Sigma_{j}$ consisting of some copies of various components of $K$ (so that $S_{j}$ is the union of $\Sigma_{j}$ and the corresponding cores of the 2-handles).

Given

$$\alpha^{\prime}\in H_{2}^{L}(W_{4};{\mathbb{Z}})\cong H_{2}^{L}(W_{3};{\mathbb{Z}})\cong H_{2}^{L}(W_{2};{\mathbb{Z}})/([S_{1}],\dots,[S_{p}]),$$

let $\langle\alpha^{\prime}\rangle$ be the set of all $\alpha\in H_{2}^{L}(W_{2};{\mathbb{Z}})\subseteq{\mathbb{Z}}^{n}$ whose equivalence class modulo $([S_{1}],\dots,[S_{p}])$ is $\alpha^{\prime}$.

Then, the skein lasagna module $\mathcal{S}_{0}^{N}(W_{4};L,\alpha^{\prime})$ is isomorphic to the quotient of the direct sum of cabled skein lasagna modules $\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)$ by the relations

(14)

$$\underline{\Psi}_{I\times\partial W_{1};\Sigma_{j}(n\bullet),\alpha^{\prime}}(v)=0,\ \ n=0,1,\dots,N-2,$$

and

(15)

$$\underline{\Psi}_{I\times\partial W_{1};\Sigma_{j}((N-1)\bullet),\alpha^{\prime}}(v)=v$$

for all $v\in\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)$ and $j=1,\dots,p$.