Strong and weak $(1,3)$ homotopies on knot projections

Abstract

Strong and weak $(1, 3)$ homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two different types of the third flat Reidemeister moves. In this paper, we introduce the cross chord number that is the minimal number of double points of chords of a chord diagram. Cross chord numbers induce a strong $(1, 3)$ invariant. We show that Hanaki's trivializing number is a weak $(1, 3)$ invariant. We give a complete classification of knot projections having trivializing number two up to the first flat Reidemeister moves using cross chord numbers and the positive resolutions of double points. Two knot projections with trivializing number two are both weak $(1, 3)$ homotopy equivalent and strong $(1, 3)$ homotopy equivalent if and only if they can be related by only the first flat Reidemeister moves. Finally, we determine the strong $(1, 3)$ homotopy equivalence class containing the trivial knot projection and other classes of knot projections.