Invariants of surfaces in smooth 4-manifolds from link homology

Abstract

We construct analogs of Khovanov-Jacobsson classes and the Rasmussen invariant for links in the boundary of any smooth oriented 4-manifold. The main tools are skein lasagna modules based on equivariant and deformed versions of $\mathfrak{gl}_N$ link homology, for which we prove non-vanishing and decomposition results. Along the way, we characterize precise technical conditions that allow a link homology theory to extend to skein lasagna 4-manifold invariants, we establish a decomposition theorem for deformed $\mathfrak{gl}_N$ skein lasagna modules, and we illustrate how Hopf link homology classes can be used to extend the functoriality of link homology theories to immersed link cobordisms.