From zero surgeries to candidates for exotic definite four-manifolds

TL;DR

This paper investigates how zero-surgery homeomorphisms can be used to generate potential exotic smooth structures on 4-manifolds, providing a systematic construction of knot pairs with identical zero surgeries and exploring their implications.

Contribution

It introduces a general construction method for knot pairs with the same zero surgeries and explores their role in producing exotic smooth structures on 4-manifolds.

Findings

Identified 5 topologically slice knots that could lead to exotic four-spheres if slice.

Developed a systematic approach to generate knot pairs with identical zero surgeries.

Explored potential exotic smooth structures on connected sums of complex projective planes.

Abstract

One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W'$. In this paper we explore how $0$-surgery homeomorphisms can be used to potentially construct exotic pairs of this form. In order to systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find $5$ topologically slice knots such that, if any of them were slice, we would obtain an exotic four-sphere. We also investigate the possibility of constructing exotic smooth structures on $\#^n \mathbb{C}P^2$ in a similar fashion.