Embedding spheres in knot traces

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

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.

• Item Type: Articles
• Additional Information: ANM is supported by NSF DMS-1902880. PF and MN gratefully acknowledge support by the SNSF Grant 181199.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Powell, Dr Mark
• Authors: Feller, P., Miller, A. N., Nagel, M., Orson, P., Powell, M., and Ray, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Compositio Mathematica
• Publisher: Cambridge University Press
• ISSN: 0010-437X
• ISSN (Online): 1570-5846
• Published Online: 20 October 2021
• Copyright Holders: Copyright © 2021 The Author(s)
• First Published: First published in Compositio Mathematica 157(10): 2242-2279
• Publisher Policy: Reproduced under a Creative Commons license