Knotting corks

TL;DR

This paper constructs infinitely many knotted cork embeddings in 4-manifolds, demonstrating their role in generating diverse exotic smooth structures, and extends the construction to plugs.

Contribution

It introduces methods to embed corks and plugs in 4-manifolds in infinitely many knotted ways, producing distinct exotic smooth structures.

Findings

Infinitely many knotted cork embeddings induce different exotic structures.

Finite disjoint cork embeddings can produce multiple distinct smooth structures.

Similar constructions are achieved for plugs.

Abstract

It is known that every exotic smooth structure on a simply connected closed 4-manifold is determined by a codimention zero compact contractible Stein submanifold and an involution on its boundary. Such a pair is called a cork. In this paper, we construct infinitely many knotted imbeddings of corks in 4-manifolds such that they induce infinitely many different exotic smooth structures. We also show that we can imbed an arbitrary finite number of corks disjointly into 4-manifolds, so that the corresponding involutions on the boundary of the contractible 4-manifolds give mutually different exotic structures. Furthermore, we construct similar examples for plugs.