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

Noboru Ito, Yusuke Takimura

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)


In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

Original languageEnglish
Article number1350085
JournalJournal of Knot Theory and its Ramifications
Issue number14
Publication statusPublished - 2013 Dec
Externally publishedYes


  • (1, 2) homotopy
  • flat Reidemeister move
  • Knot projection
  • weak (1, 3) homotopy

ASJC Scopus subject areas

  • Algebra and Number Theory


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