ScalingStacks

Proof. First, note that the addition of 4-handles does not affect the skein lasagna module, in view of Proposition 3.4. Thus, we can consider $W_{3}$ instead of $W_{4}$.

The skein lasagna module of $L$ viewed in the boundary of $\partial W$ is given by $\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)$ according to Theorem 3.2. When we add a 3-handle, we divide by the relations

(16)

$$\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}(v\otimes X^{n})=\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}(v\otimes X^{n}),$$

as proved in Theorem 3.7. In terms of the identifications $\Phi$ from Theorem 3.2, the left hand side of (16) is given by Equation (11), and the right hand side by Equation (13). We thus get relations of the form (14) and (15). The generalization to multiple 3-handles is straightforward. ∎