Origami edge-paths in the curve graph

TL;DR

This paper introduces the concept of origami edge-paths in the curve graph of a surface, proving their existence for any origami pair of curves and exploring shortest paths.

Contribution

It establishes the existence of origami edge-paths connecting origami pairs of curves and characterizes their properties, advancing understanding of curve interactions on origami surfaces.

Findings

Existence of origami edge-paths for any origami pair of curves.

Any such path consists of curves intersecting exactly once.

All pairs along the path are coherent and form origami pairs.

Abstract

An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ 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, $ (\alpha,\beta)$, 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, $\alpha =\alpha_0, \alpha_1, \alpha_2, \cdots, \alpha_n = \beta $, such that: $\alpha_i$ intersects $\alpha_{i+1}$ at exactly once; any pair $(\alpha_i, \alpha_j)$ is coherent; and thus, any filling pair, $(\alpha_i, \alpha_j)$, is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.