Lemma 2.1 (0NFP) · ScalingStacks

0NFP

Lemma 2.1. Let $W$ and $L$ be as above, and fix balls $R_{1},\dots,R_{n}$ , one in each connected component of $W$ . Then, the equivalence relation defining $\mathcal{S}_{0}^{N}(W;L)$ can be alternatively be described as the transitive and linear closure of the following relation:

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Linear combinations of lasagna fillings are set to be multilinear in the labels $v_{i}$ ;
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Lasagna fillings that are isotopic rel $\partial W$ are set to be equivalent;
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Two lasagna fillings are also set to be equivalent if they differ as in (b) above, where the input ball $B_{i}$ is one of the chosen balls $R_{1},\dots,R_{n}$ .

0NFQ

Proof. If $F_{1}$ and $F_{2}$ are equivalent as in the lemma, let us show that they are equivalent as in the definition of the skein lasagna module. The only new relation is the isotopy, which can be thought of as a particular instance of (b), where $B_{1}$ is replaced by a slightly smaller ball with the same decoration (and $F_{3}$ is a product cobordism).

Conversely, if $F_{1}$ and $F_{2}$ are equivalent as in the definition of the skein lasagna module, we only have to consider the case when they are related by (b). We can then isotope $B_{i}$ to turn it into the ball $R_{j}$ in the same connected component, and view (b) as a combination of the moves in the lemma. ∎