Distinguishing exotic $R^4$'s with Heegaard Floer homology

Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to $R^4$. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic $R^4$’s made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise nondiffeomorphic exotic $R^4$’s. Our main tool is Gadgil’s end Floer homology. This is joint work with Sean Eli and Tye Lidman.