Handle decompositions and the 1-dimensional inputs skein lasagna module

Abstract

We establish handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over $\mathbb{Q}$, defined recently by Ren, Wedrich, Willis, Zhang, and the second author. For a $4$-manifold built out of $1$- and $2$-handles, the invariant can be computed in terms of a cabled colimit of Rozansky-Willis homologies, modulo a new relation which we call the lasso relation. We then present some explicit calculations for disk bundles over $S^{2}$, as well as a partial vanishing result for $4$-manifolds of the form $\Sigma_{g}\times D^{2}$, $g\geq 1$.