Seifert vs. slice genera of knots in twist families and a characterization of braid axes

Kenneth L. Baker, Kimihiko Motegi

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Twisting a knot (Formula presented.) in (Formula presented.) along a disjoint unknot (Formula presented.) produces a twist family of knots (Formula presented.) indexed by the integers. We prove that if the ratio of the Seifert genus to the slice genus for knots in a twist family limits to 1, then the winding number of (Formula presented.) about (Formula presented.) equals either zero or the wrapping number. As a key application, if (Formula presented.) or the mirror twist family (Formula presented.) contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that (Formula presented.) is a braid axis of (Formula presented.) if and only if both (Formula presented.) and (Formula presented.) each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for (Formula presented.) to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker–Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman–Moore.

Original languageEnglish (US)
Pages (from-to)1493-1530
Number of pages38
JournalProceedings of the London Mathematical Society
Volume119
Issue number6
DOIs
StatePublished - 2019

Keywords

  • 57M25
  • 57M27 (primary)
  • 57R17
  • 57R58 (secondary)

ASJC Scopus subject areas

  • Mathematics(all)

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