Noncharacterizing slopes for hyperbolic knots

Kenneth Baker, Kimihiko Motegi

Research output: Contribution to journalArticle

1 Citation (Scopus)

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

Hyperbolic Knot
Slope
Knot
Unknot
Surgery
Meridian
L-space
Homeomorphic

Keywords

  • Characterizing slope
  • Dehn surgery

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Noncharacterizing slopes for hyperbolic knots. / Baker, Kenneth; Motegi, Kimihiko.

In: Algebraic and Geometric Topology, Vol. 18, No. 3, 09.04.2018, p. 1461-1480.

Research output: Contribution to journalArticle

Baker, Kenneth ; Motegi, Kimihiko. / Noncharacterizing slopes for hyperbolic knots. In: Algebraic and Geometric Topology. 2018 ; Vol. 18, No. 3. pp. 1461-1480.
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