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Embedding spheres in knot traces

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

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Authors

Peter Feller

Allison N. Miller

Matthias Nagel

Patrick Orson

Mark Powell

Arunima Ray



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
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

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Publisher Licence URL
http://creativecommons.org/licenses/by-nc/4.0/

Copyright Statement
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial licence (http://creativecommons.org/licenses/by-nc/4.0), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the original article is properly cited. Written permission must be obtained prior to any commercial use. Compositio Mathematica is © Foundation Compositio Mathematica.




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