  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.
