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

TL;DR

This paper introduces an algorithm to count minimally intersecting filling coherent pairs of curves on genus g surfaces and establishes a new upper bound for their number using the Ménage Problem, linking origami structures and intersection theory.

Contribution

It presents a novel algorithm for counting minimal filling pairs and derives an upper bound using combinatorial methods related to origami structures.

Findings

Developed an algorithm to count minimal filling pairs.

Established a new upper bound using the Ménage Problem.

Linked origami structures with intersection properties of curves.

Abstract

Let $S_g$ denoting the genus $g$ closed orientable surface. An {\em origami} (or flat structure) on $S_g$ 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. Coherent filling pairs of simple closed curves, $(\alpha,\beta)$ in $S_g$ are pairs for which their minimal intersection is equal to their algebraic intersection. And, a minimally intersecting filling of $(\alpha,\beta)$ in $S_g$ is a pair whose intersection number is the minimal among all filling pairs of $S_g$. A coherent pair of curves is naturally associated with an origami on $S_g$, and a minimally intersecting filling coherent pair of curves has the smallest number of squares in all origamis on $S_g$. Our main result introduce an algorithm to count the numbers of minimal filling pairs on $S_g$, and establish a new upper bound of this count using M\'enage Problem.