Exotically knotted surfaces and the Bar-Natan lasagna module

Abstract: In this talk, we define and state properties about a homological skein invariant constructed from Bar-Natan's deformation of Khovanov homology. We extend the notion of $H$-torsion order for Bar-Natan homology and corresponding results about internal stabilization distances of exotically knotted surfaces to the 4-manifold setting through this construction. In particular, we examine gluing maps for this invariant corresponding to connect-sums of 4-manifolds, and we use these maps in conjunction with results of Hayden to produce exotic pairs of knotted surfaces in 4-manifolds other than the 4-ball. Furthermore, we show that these surfaces do not become smoothly isotopic rel boundary after a single internal stabilization.