Skein lasagna modules and Rozansky-Willis homology

In this talk I will describe joint work with Ian Sullivan, where we use properties of categorified projectors to prove that the Khovanov skein lasagna module of $S^2 \times S^2$ is trivial. Along the way, we will discover a relationship between the skein lasagna module of $S^2 \times D^2$ with a link $L$ in the boundary and the Rozansky-Willis homology of $L$ inside $S^2 \times S^1$. This isomorphism is used in recent joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis, where we define a new version of $gl_2$ skein lasagna modules with 1-dimensional inputs.