Geometry and Topology Seminar

In 2018, Morrison, Walker, and Wedrich's skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining an isomorphic object called cabled Khovanov-Rozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4-manifold.

One especially important conjecture states that the skein lasagna module of $S^2\times S^2$ is 0 (or infinite-dimensional). This is a necessary condition for the invariant to be able to detect exotic 4-manifold pairs. In this project, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan's category. This allows us to compute significant portions of the skein lasagna module of $S^2 \times S^2$, and relate the remainder to the Rozansky-Willis invariant of links in $S^2 \times S^1$. Our local techniques also allow for computations of the skein lasagna invariant for other 4-manifolds whose Kirby diagram contains a 0-framed unknot component.

This is joint upcoming work with Ian Sullivan (UC Davis).