Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link

Thomas Fleming, Tetsuo Shibuya, Tatsuya Tsukamoto, Akira Yasuhara

Research output: Contribution to journalArticle

Abstract

Link-homotopy and self Δ-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Δ-equivalent) to a trivial link. We study link-homotopy and self Δ-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Δ-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Δ-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Δ-equivalence and concordance.

Original languageEnglish
Pages (from-to)1215-1227
Number of pages13
JournalTopology and its Applications
Volume157
Issue number7
DOIs
Publication statusPublished - 2010 May 1
Externally publishedYes

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Keywords

  • Link-homotopy
  • Milnor invariants
  • Self Δ-equivalence

ASJC Scopus subject areas

  • Geometry and Topology

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