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.