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Geometry and Topology Seminar

Friday, January 19, 2024
4:00pm to 5:00pm
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Linde Hall 187
Kirby and the Skein Lasagna Module of $S^2 \times S^2$
Melissa Zhang, Department of Mathematics, UC Davis,

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).

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected] or visit