Khovanov skein lasagna modules with 1-dimensional inputs

Skein lasagna modules (with 0-dimensional inputs) are 4-manifold invariants introduced by Morrison-Walker-Wedrich. In this context, a skein is a properly embedded surface in a 4-manifold minus a disjoint union of 4-balls, and the lasagna comes from a TQFT for links in S^3 (satisfying mild conditions). In this talk, we introduce skein lasagna modules with 1-dimensional inputs, where a skein is a properly embedded surface in a 4-manifold minus a tubular neighborhood of an embedded graph, and the lasagna comes from a TQFT for links in \#(S^1\times S^2) and link cobordisms between them in a particular class of 4-manifolds. We present (a subset of) four key ingredients which enable us to show that the Khovanov homology of links in \#(S^1\times S^2), as defined by Rozansky and Willis, has excellent functoriality properties sufficient to supply the lasagna input TQFT. The ingredients are: 

0. A lasagna interpretation of Rozansky-Willis homology by Sullivan-Zhang; 

1. A calculation of the diffeomorphism group (rel boundary, modulo local diffeomorphisms) of the complement of any graph in S^4; 

2. A Khovanov lasagna naturality property of the Gluck twist operation; 

3. A naturality result for the Khovanov-Rozansky gl_2-homology of singular gl_2-foams between gl_2-webs in S^3. 

This is joint work with I. Sullivan, P. Wedrich, M. Willis, M. Zhang.