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

Noboru Ito, Yusuke Takimura

研究成果: Article

2 引用 (Scopus)

抄録

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.

元の言語English
記事番号1550069
ジャーナルInternational Journal of Mathematics
26
発行部数9
DOI
出版物ステータスPublished - 2015 8 29
外部発表Yes

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Knot
Projection
Equivalence class
Homotopy
Homotopy Equivalence
Invariant
Equivalence relation
Immersion
Replacement
Circle

ASJC Scopus subject areas

  • Mathematics(all)

これを引用

Strong and weak (1, 2, 3) homotopies on knot projections. / Ito, Noboru; Takimura, Yusuke.

:: International Journal of Mathematics, 巻 26, 番号 9, 1550069, 29.08.2015.

研究成果: Article

Ito, Noboru ; Takimura, Yusuke. / Strong and weak (1, 2, 3) homotopies on knot projections. :: International Journal of Mathematics. 2015 ; 巻 26, 番号 9.
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