Any nontrivial knot projection with no triple chords has a monogon or a bigon

TL;DR

This paper proves that any nontrivial knot projection without triple chords can be simplified to a circle through a sequence of specific Reidemeister moves that reduce monogons or bigons.

Contribution

It establishes a new simplification process for knot projections lacking triple chords using moves that decrease monogons or strong bigons.

Findings

Any such knot projection can be reduced to a simple circle.

The reduction process involves only flat Reidemeister moves decreasing monogons or strong bigons.

The result links chord diagram properties to knot projection simplification.

Abstract

A generic immersion of a circle into a $2$-sphere is often studied as a projection of a knot; it is called a knot projection. A chord diagram is a configuration of paired points on a circle; traditionally, the two points of each pair are connected by a chord. A triple chord is a chord diagram consisting of three chords, each of which intersects the other chords. Every knot projection obtains a chord diagram in which every pair of points corresponds to the inverse image of a double point. In this paper, we show that for any knot projection $P$, if its chord diagram contains no triple chord, then there exists a finite sequence from $P$ to a simple closed curve such that the sequence consists of flat Reidemeister moves, each of which decreases $1$-gons or strong $2$-gons, where a strong $2$-gon is a $2$-gon oriented by an orientation of $P$.