Origamis associated to minimally intersecting filling pairs

TL;DR

This paper constructs a large number of mapping class group orbits of curve pairs on surfaces, each giving rise to origamis, and explores their properties with experimental data and conjectures.

Contribution

It improves previous results by constructing factorially-many such orbits, revealing new origami structures associated with minimally intersecting filling pairs.

Findings

Constructed factorially-many mapping class group orbits

Each orbit corresponds to an origami with specific intersection properties

Collected experimental data on $SL(2, \\mathbb{Z})$-orbits and proposed conjectures

Abstract

Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Using different techniques, we improve on this result by constructing factorially-many (again in $g$) such orbits. These new orbits are chosen so that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami. We collect some rudimentary experimental data on the corresponding $SL(2, \mathbb{Z})$-orbits and suggest further study and conjectures.