Strong and weak (1, 2, 3) homotopies on knot projections

Noboru Ito, Yusuke Takimura

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first nontrivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e. Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible.

Original languageEnglish
Article number1550069
JournalInternational Journal of Mathematics
Issue number9
Publication statusPublished - 2015 Aug 29
Externally publishedYes


  • Generic spherical curves
  • homotopy
  • knot projections
  • Reidemeister moves

ASJC Scopus subject areas

  • Mathematics(all)


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