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Concordance, Crossing Changes, and Knots in Homology Spheres

Part of: Low-dimensional topology

Published online by Cambridge University Press:  16 December 2019

Christopher W. Davis
Christopher W. Davis*
Affiliation:
Department of Mathematics, University of Wisconsin–Eau Claire, USA Email: daviscw@uwec.edu URL: www.uwec.edu/daviscw
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Abstract

Any knot in $S^{3}$ can be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot that is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere.

MSC classification

Primary: 57M25: Knots and links in $S^3$
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 744 - 754
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Copyright
© Canadian Mathematical Society 2019

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