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Description

Khovanov skein lasagna modules describe a smooth 4-manifold in terms of the surfaces embedded within; these surfaces have 0-dimensional singularities, thought of as "inputs", in the following way: the boundary of a $B^4$ neighborhood of each singularity is an $S^3$ which the surface intersects at a link; Khovanov homology is used here to label these links at these singularities. This invariant has been used to study questions related to the Smooth Poincare Conjecture in Dimension 4; the focus of these questions is on "exotic behavior" in 4-manifolds, i.e. the difference between being homeomorphic and diffeomorphic. While the original skein lasagna modules with 0-dimensional inputs can in some cases be computed for 4-manifolds built with only 0-, 2-, and 4-handles, it is currently not computable for 4-manifolds with 1-handles. Nevertheless, many of our known examples of exotic pairs of 4-manifolds involve 1-handles, and so there has been much interest in either finding a `1-handle formula’ or developing a 1-handle-friendly version of skein lasagna modules. In this talk I will describe joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis, where we define a new version of skein lasagna modules from $gl_2$ link homology, with 1-dimensional inputs, which is more amenable to 4-manifolds with 1-handles. The strategy is to use an isomorphism discovered in previous joint work with Ian Sullivan, where we related the skein lasagna module of $S^2 \times D^2$ to Rozansky-Willis homology, a version of Khovanov homology for links in connected sums of $S^2 \times S^1$. I will begin with introductions to the relevant ingredients, such as Khovanov homology, functoriality, skein lasagna modules, and the categorified projectors that are used in Rozansky-Willis homology.