Smooth and topological almost concordance

Nagel, Matthias; Orson, Patrick; Park, JungHwan; Powell, Mark

doi:10.1093/imrn/rnx338

Smooth and topological almost concordance

Nagel, Matthias; Orson, Patrick; Park, JungHwan; Powell, Mark

Authors

Matthias Nagel

Patrick Orson

JungHwan Park

Mark Powell

Abstract

We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Y × [0, 1] and the action of the concordance group of knots in S3 that ties in local knots. We prove that the trivial free homotopy class in every 3-manifold other than the 3-sphere contains an infinite family of knots, all topologically concordant, but not smoothly almost concordant to one another. Then, in every lens space and for every free homotopy class, we find a pair of topologically concordant but not smoothly almost concordant knots. Finally, as a topological counterpoint to these results, we show that in every lens space every free homotopy class contains infinitely many topological almost concordance classes.

Citation

Nagel, M., Orson, P., Park, J., & Powell, M. (2019). Smooth and topological almost concordance. International Mathematics Research Notices, 2019(23), 7324-7355. https://doi.org/10.1093/imrn/rnx338

Journal Article Type	Article
Acceptance Date	Dec 29, 2017
Online Publication Date	Feb 5, 2018
Publication Date	Dec 31, 2019
Deposit Date	Jan 4, 2018
Publicly Available Date	Feb 5, 2019
Journal	International Mathematics Research Notices
Print ISSN	1073-7928
Electronic ISSN	1687-0247
Publisher	Oxford University Press
Peer Reviewed	Peer Reviewed
Volume	2019
Issue	23
Pages	7324-7355
DOI	https://doi.org/10.1093/imrn/rnx338
Related Public URLs	https://arxiv.org/abs/1707.01147

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Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Nagel, Matthias, Orson, Patrick, Park, JungHwan & Powell, Mark (2018). Smooth and topological almost concordance. International Mathematics Research Notices is available online at: https://doi.org/10.1093/imrn/rnx338.