Noncharacterizing slopes for hyperbolic knots

Baker, Kenneth; Motegi, Kimihiko

doi:10.2140/agt.2018.18.1461

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Noncharacterizing slopes for hyperbolic knots

Kenneth L Baker and Kimihiko Motegi

Algebraic & Geometric Topology 18 (2018) 1461–1480

DOI: 10.2140/agt.2018.18.1461

Abstract

A nontrivial slope $r$ on a knot $K$ in $S^{3}$ is called a characterizing slope if whenever the result of $r$ –surgery on a knot $K^{'}$ is orientation-preservingly homeomorphic to the result of $r$ –surgery on $K$ , then $K^{'}$ is isotopic to $K$ . Ni and Zhang ask: for any hyperbolic knot $K$ , is a slope $r = p ∕ q$ with $| p | + | q |$ sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot $c$ so that $(0, 0)$ –surgery on $K \cup c$ results in $S^{3}$ and $c$ is not a meridian of $K$ , then $K$ has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot $8_{6}$ has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot $c$ .

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Keywords

Dehn surgery, characterizing slope

Mathematical Subject Classification 2010

Primary: 57M25

References

Publication

Received: 21 November 2016

Revised: 15 May 2017

Accepted: 13 August 2017

Published: 3 April 2018

Authors

Kenneth L Baker
Department of Mathematics University of Miami Coral Gables, FL United States

Kimihiko Motegi
Department of Mathematics Nihon University Tokyo Japan

Volume 18, issue 3 (2018)