Embedding spheres in knot traces

Feller, Peter; Miller, Allison N.; Nagel, Matthias; Orson, Patrick; Powell, Mark; Ray, Arunima

Abstract

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.

Citation

Feller, P., Miller, A. N., Nagel, M., Orson, P., Powell, M., & Ray, A. (2021). Embedding spheres in knot traces. Compositio Mathematica, 157(10), 2242-2279. https://doi.org/10.1112/s0010437x21007508

• Journal Article Type: Article
• Acceptance Date: Apr 8, 2021
• Online Publication Date: Oct 20, 2021
• Publication Date: 2021
• Deposit Date: Apr 10, 2021
• Publicly Available Date: Oct 25, 2021
• Journal: Compositio Mathematica
• Print ISSN: 0010-437X
• Electronic ISSN: 1570-5846
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 157
• Issue: 10
• Pages: 2242-2279
• DOI: https://doi.org/10.1112/s0010437x21007508
• Public URL: https://durham-repository.worktribe.com/output/1277756
• Publisher URL: https://www.cambridge.org/core/journals/compositio-mathematica/article/embedding-spheres-in-knot-traces/60143F71ABB920CEAEF0C2AE858374F7
• Related Public URLs: https://arxiv.org/abs/2004.04204