Embedding spheres in knot traces. (20th October 2021)

• Record Type: Journal Article
• Authors: Feller, Peter; Miller, Allison N.; Nagel, Matthias; Orson, Patrick; Powell, Mark; Ray, Arunima
• Is Part Of: Compositio mathematica. Volume 157:Number 10(2021)
• Page Start: 2242
• Page End: 2279
• Publication Date: 2021-10-20
• Subjects: shake slice -- locally flat embedding -- Arf invariant -- Tristram–Levine signatures -- Alexander polynomial; 57K40 -- 57K10 -- 57N35 -- 57N70 -- 57R67; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007508
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract: 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.