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 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 inputs. We touch upon the three key ingredients of the proof: a lasagna interpretation of Rozansky-Willis homology by Sullivan-Zhang; Gabai's $4$-dimensional lightbulb theorem; and certain Khovanov lasagna naturality properties of the Gluck twist operation. This is joint work with I. Sullivan, P. Wedrich, M. Willis, M. Zhang.