Gompf’s Cork and Heegaard Floer Homology

Gompf showed that for K in a certain family of double-twist knots, the swallow-follow operation makes $1/n$-surgery on $K # -K$ into a cork boundary. We derive a general Floer-theoretic condition on K under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf’s method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms.