Dehn surgeries and negative-definite four-manifolds

TL;DR

This paper investigates which Dehn surgeries on knots in the three-sphere bound negative-definite four-manifolds, introducing an invariant m(K) that determines this and is computed for torus knots.

Contribution

It introduces the invariant m(K) for knots, linking Dehn surgery properties to smooth concordance, and computes it for specific classes like torus knots.

Findings

m(K) determines when Dehn surgeries bound negative-definite four-manifolds

m(K) is a smooth concordance invariant

Computed m(K) for torus knots

Abstract

Given a knot K in the three-sphere, we address the question: which Dehn surgeries on K bound negative-definite four-manifolds? We show that the answer depends on a number m(K), which is a smooth concordance invariant. We study the properties of this invariant, and compute it for torus knots.