ScalingStacks

Definition 5.4. Let $W$ be a smooth oriented 4-manifold and $L\subset\partial W$ a link. Then we define the bigraded abelian group

$$\mathcal{S}^{N}_{0}(W;L)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}{\mathbb{Z}}\{\text{lasagna fillings{} }F\text{ of }W\text{ with boundary }L\}/\sim$$

where $\sim$ is the transitive and linear closure of the relation on lasagna fillings for which $F_{1}\sim F_{2}$ if $F_{1}$ has an input ball $B_{1}$ with label $v_{1}$, and $F_{2}$ can be obtained from $F_{1}$ by replacing $B_{1}$ with a third lasagna filling $F_{3}$ of a 4-ball such that $v_{1}=\mathrm{KhR}_{N}(F_{3})$, followed by an isotopy rel boundary. This is illustrated in Figure 3.