Embedding spheres in knot traces

Abstract

AbstractThe trace of the$n$-framed surgery on a knot in$S^{3}$is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded$2$-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable$3$-dimensional knot invariants. For each$n$, this provides conditions that imply a knot is topologically$n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.