#### Volume 18, issue 3 (2018)

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Noncharacterizing slopes for hyperbolic knots

### Kenneth L Baker and Kimihiko Motegi

Algebraic & Geometric Topology 18 (2018) 1461–1480
##### Abstract

A nontrivial slope $r$ on a knot $K$ in ${S}^{3}$ is called a characterizing slope if whenever the result of $r\phantom{\rule{-0.17em}{0ex}}$–surgery on a knot ${K}^{\prime }$ is orientation-preservingly homeomorphic to the result of $r\phantom{\rule{-0.17em}{0ex}}$–surgery on $K$, then ${K}^{\prime }$ 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 $\left(0,0\right)$–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 $c$.

##### Keywords
Dehn surgery, characterizing slope
Primary: 57M25
##### 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