Infinite order corks via handle diagrams

Abstract

The author recently proved the existence of an infinite order cork: a compact, contractible submanifold $C$ of a 4–manifold and an infinite order diffeomorphism $f$ of $\partial C$ such that cutting out $C$ and regluing it by distinct powers of $f$ yields pairwise nondiffeomorphic manifolds. The present paper exhibits the first handle diagrams of this phenomenon, by translating the earlier proof into this language (for each of the infinitely many corks arising in the first paper). The cork twists in these papers are twists on incompressible tori. We give conditions guaranteeing that such twists do not change the diffeomorphism type of a 4–manifold, partially answering a question from the original paper. We also show that the “$\delta$–moves” recently introduced by Akbulut are essentially equivalent to torus twists.