Noncharacterizing slopes for hyperbolic knots

Kenneth Baker, Kimihiko Motegi

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1461-1480
Number of pages20
JournalAlgebraic and Geometric Topology
Volume18
Issue number3
DOIs
StatePublished - Apr 9 2018

    Fingerprint

Keywords

  • Characterizing slope
  • Dehn surgery

ASJC Scopus subject areas

  • Geometry and Topology

Cite this