Any Nontrivial Knot Projection with No Triple Chords Has a Monogon or a Bigon - Ito, Noboru; Takimura, Yusuke (Topology Proceedings, 2021)
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 $P$.