TY - JOUR

T1 - Noncharacterizing slopes for hyperbolic knots

AU - Baker, Kenneth L.

AU - Motegi, Kimihiko

N1 - Funding Information:
Baker was partially supported by a grant from the Simons Foundation (#209184 to Kenneth L Baker). Motegi was partially supported by JSPS KAKENHI Grant Number JP26400099 and Joint Research Grant of Institute of Natural Sciences at Nihon University for 2016.

PY - 2018/4/9

Y1 - 2018/4/9

N2 - 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=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∪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.

AB - 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=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∪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.

KW - Characterizing slope

KW - Dehn surgery

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U2 - 10.2140/agt.2018.18.1461

DO - 10.2140/agt.2018.18.1461

M3 - Article

AN - SCOPUS:85045224321

VL - 18

SP - 1461

EP - 1480

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 3

ER -