From zero surgeries to candidates for exotic definite 4-manifolds

One strategy for distinguishing smooth structures on closed 4-manifolds is to produce a knot K$K$ in S3$S<^>3$ that is slice in one smooth filling W$W$ of S3$S<^>3$ but not slice in some homeomorphic smooth filling W & PRIME;$W<^>{\prime }$. In this paper, we explore how 0-surgery homeomorphisms can be used to potentially construct exotic pairs of this form. 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 five topologically slice knots such that, if any of them were slice, we would obtain an exotic 4-sphere. We also investigate the possibility of constructing exotic smooth structures on #nCP2$\#<^>n \mathbb {CP}<^>2$ in a similar fashion.