The trace of the n-framed surgery on a knot in S³ 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.
shake slice, locally flat embedding, Arf invariant, Tristram–Levine signatures, Alexander polynomial
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P. Feller, Allison N. Miller, M. Nagel, P. Orson, M. Powell, and A. Ray.
"Embedding Spheres In Knot Traces".