ScalingStacks

Definition 5.1. A lasagna algebra ${\mathcal{L}}$ consists of

• •

  for each link $L$ in a 3-sphere $S$, a (bigraded) abelian group ${\mathcal{L}}(S,L)$, which depends functorially on the pair $(S,L)$,

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  for each lasagna diagram $D$, which, by definition, consists of a 4-ball $B$, with a finite collection of disjoint 4-balls $B_{i}$ removed from the interior, with boundary components $S$ (on the outside) and $S_{i}$ (the boundaries of the removed interior balls $B_{i}$), and properly embedded framed oriented surface $\Sigma$ in the complementary region, meeting the boundary spheres in links $L$ and $L_{i}$ (see Figure 2), a (homogeneous) homomorphism

  $${\mathcal{L}}(D):\bigotimes_{i}{\mathcal{L}}(S_{i},L_{i})\to{\mathcal{L}}(S,L),$$

  such that

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  surfaces $\Sigma$ and $\Sigma^{\prime}$ which are isotopic rel boundary induce identical homomorphisms,

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  if $f:D\to D^{\prime}$ is a diffeomorphism between lasagna diagrams, then the square

  commutes,

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  a ‘radial’ surface $L\times{[0,1]}\subset S\times{[0,1]}$ induces the identity map ${\mathcal{L}}(S,L)\to{\mathcal{L}}(S,L)$ (or more precisely, mapping cylinders of diffeomorphisms induce the same map specified for that diffeomorphism by functoriality),

• •

  gluing of a ‘smaller’ lasagna diagram into one of the removed balls of a ‘larger’ lasagna diagram (with compatible boundaries) to obtain a single lasagna diagram is compatible with the corresponding composition of homomorphisms.