A construction of minimal coherent filling pairs

TL;DR

This paper presents a geometric method to construct minimal intersecting coherent filling pairs on genus g surfaces, starting from pairs on a torus, and explores their connection to square-tiled surfaces.

Contribution

It introduces a simple geometric procedure for constructing minimal coherent filling pairs on higher genus surfaces from torus pairs, linking to origami structures.

Findings

Constructed minimal intersecting coherent filling pairs for g ≥ 3

Established a correspondence between these pairs and square-tiled surfaces

Discussed origami surfaces derived from the construction

Abstract

Let $S_g$ denote the genus $g$ closed orientable surface. A \emph{coherent filling pair} of simple closed curves, $(\alpha,\beta)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its algebraic intersection number. A \emph{minimally intersecting} filling pair, $(\alpha,\beta)$ in $S_g$, is one whose intersection number is the minimal among all filling pairs of $S_g$. In this paper, we give a simple geometric procedure for constructing minimal intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or {\em origamis}, and we discuss the origami obtained from the construction.