Origami edge-paths in the curve graph

Abstract

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 intersectionthey 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 = α 0 , α 1 , α 2 , • • • , α n = β, such that: α i intersects α i+1 at 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.

An upper bound of the numbers of minimally intersecting filling coherent pairs

“…Theorem 3 (Theorem 1.1 of [3]). A coherent filling pair of curves(origami pair of curves) naturally corresponds to an origami on S g .…”