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Noncharacterizing slopes for hyperbolic knots

Kenneth L Baker and Kimihiko Motegi

Algebraic & Geometric Topology 18 (2018) 1461–1480

A nontrivial slope r on a knot K in S3 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 = pq 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 c results in S3 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 86 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.

Dehn surgery, characterizing slope
Mathematical Subject Classification 2010
Primary: 57M25
Received: 21 November 2016
Revised: 15 May 2017
Accepted: 13 August 2017
Published: 3 April 2018
Kenneth L Baker
Department of Mathematics
University of Miami
Coral Gables, FL
United States
Kimihiko Motegi
Department of Mathematics
Nihon University