## Journal of the Mathematical Society of Japan

### Triple chords and strong (1, 2) homotopy

#### Abstract

A triple chord $\rlap{\ominus}\otimes$ is a sub-diagram of a chord diagram that consists of a circle and finitely many chords connecting the preimages for every double point on a spherical curve. This paper describes some relationships between the number of triple chords and an equivalence relation called strong (1, 2) homotopy, which consists of the first and one kind of the second Reidemeister moves involving inverse self-tangency if the curve is given any orientation. We show that a knot projection is trivialized by strong (1, 2) homotopy, if it is a simple closed curve or a prime knot projection without $1$- and $2$-gons whose chord diagram does not contain any triple chords. We also discuss the relation between Shimizu's reductivity and triple chords.

#### Article information

Source
J. Math. Soc. Japan, Volume 68, Number 2 (2016), 637-651.

Dates
First available in Project Euclid: 15 April 2016

https://projecteuclid.org/euclid.jmsj/1460727373

Digital Object Identifier
doi:10.2969/jmsj/06820637

Mathematical Reviews number (MathSciNet)
MR3488138

Zentralblatt MATH identifier
1341.57003

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57Q35: Embeddings and immersions

#### Citation

ITO, Noboru; TAKIMURA, Yusuke. Triple chords and strong (1, 2) homotopy. J. Math. Soc. Japan 68 (2016), no. 2, 637--651. doi:10.2969/jmsj/06820637. https://projecteuclid.org/euclid.jmsj/1460727373

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