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Embedding spheres in knot traces
- Part of
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 10 / October 2021
- Published online by Cambridge University Press:
- 20 October 2021, pp. 2242-2279
- Print publication:
- October 2021
-
- Article
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The 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.
