Whitney tower concordance and knots in homology spheres

Davis, Christopher W

doi:10.2140/agt.2025.25.3503

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Whitney tower concordance and knots in homology spheres

Christopher W Davis

Algebraic & Geometric Topology 25 (2025) 3503–3521

DOI: 10.2140/agt.2025.25.3503

Abstract

A Levine (Forum Math. Sigma 4 (2016) art. id. e34) proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in $S^{3}$ , where one allows for concordances in homology cobordisms. Since then subsequent works due to Hom, Levine and Lidman (Duke Math. J. 171 (2022) 3089–3131) and Zhou (J. Topol. 14 (2021) 1369–1395) have strengthened this result showing that there are many knots in homology spheres which are not smoothly concordant to knots in $S^{3}$ . In this paper we present evidence that the opposite is true topologically. We study the Whitney tower filtration of concordance (Ann. of Math. 157 (2003) 433–519) and prove that modulo any term in this filtration every knot (or link) in a homology sphere is equivalent to a knot (or link) in $S^{3}$ . As an application we recover a result of the author (J. Topol. 13 (2020) 343–355), namely that the solvable filtration similarly fails to distinguish links in homology spheres from links in $S^{3}$ .

Keywords

knots, links, concordance, $4$-manifolds, Whitney tower

Mathematical Subject Classification

Primary: 57K10, 57N70

References

Publication

Received: 3 April 2023

Revised: 15 March 2024

Accepted: 16 September 2024

Published: 1 October 2025

Authors

Christopher W Davis
Department of Mathematics University of Wisconsin–Eau Claire Eau Claire, WI United States

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Volume 25, issue 6 (2025)