Embedding spheres in knot traces

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