UC Davis

Many of the most well-known open problems in geometric topology involve the behavior of exotic smooth structures in dimension 4. One class of open problems in this field are the Wall-type stabilization problems, which ask about the behavior of exotic 4-manifolds under specified topological operations. For a chosen link homology theory, the associated skein lasagna module [MWW22] is an invariant of smooth 4-manifolds with robust gluing properties [MWW23]. We employ the skein lasagna module construction for Khovanov homology and study the effects of various forms of stabilization on these smooth 4-manifold invariants. Through the computations of these invariants for 4-manifolds relevant to external stabilization, we establish an isomorphism between Rozansky-Willis homology groups and skein lasagna modules of #k (S 2 × D2 ) with various null-homologous links in the boundary. Furthermore, using skein-theoretic 4-manifold invariants defined in [MWW24], we show that skein lasagna modules are capable of distinguishing pairs of exotically knotted surfaces even after an internal stabilization.