Knot homotopy in subspaces of the 3-sphere

Yuya Koda, Makoto Ozawa

Research output: Contribution to journalArticlepeer-review


We discuss an extrinsic property of knots in a 3-subspace of the 3-sphere S3 to characterize how the subspace is embedded in S3. Specifically, weshow that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e., regular neighborhoods of embedded graphs, where a knot in a 3-subspace of S3 is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in S3. To show this, we discuss the relation between certain group-theoretic and homotopic properties of knots in a compact 3-manifold, which can be of independent interest. Further, using the notion of transient knots, we define an integer-valued invariant of knots in S3 that we call the transient number. We then show that the union of the sets of knots of unknotting number one and tunnel number one is a proper subset of the set of knots of transient number one.

Original languageEnglish
Pages (from-to)389-414
Number of pages26
JournalPacific Journal of Mathematics
Issue number2
Publication statusPublished - 2016 Jun 1
Externally publishedYes


  • Homotopies
  • Knots
  • Persistent
  • Submanifolds of the 3-sphere
  • Transient

ASJC Scopus subject areas

  • Mathematics(all)


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