Origami edge-paths in the curve graph

An origami (or flat structure) on a closed oriented surface, S_g, of genus g≥2 is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main objects of study in this note are origami pairs of curves—filling pairs of simple closed curves, (α,β), in S_g such that their minimal intersection is equal to their algebraic intersection—they are coherent. An origami pair of curves is naturally associated with an origami on S_g. Our main result establishes that for any origami pair of curves there exists an origami edge-path, a sequence of curves, α=α₀,α₁,α₂,⋯,α_n=β, such that: α_i intersects α_i+1 exactly once; any pair (α_i,α_j) is coherent; and thus, any filling pair, (α_i,α_j), is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.