Triple chords and strong (1, 2) homotopy

TL;DR

This paper investigates the relationship between triple chords in chord diagrams and a specific knot equivalence called strong (1, 2) homotopy, revealing conditions under which prime knot projections become trivial under this relation.

Contribution

It establishes new connections between triple chords and strong (1, 2) homotopy, including criteria for trivializing prime knot projections and links to Shimizu's reductivity.

Findings

Prime knot projections without triple chords are trivialized by strong (1, 2) homotopy.

Prime knot projections without 1- and 2-gons are trivialized if their chord diagrams lack triple chords.

The paper relates triple chords to Shimizu's reductivity in knot theory.

Abstract

A triple chord 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, and it has exactly three chords giving the triple intersection. 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 prime 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.