Non-planarity of $\text{SL}(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$

TL;DR

This paper investigates the non-planarity of graphs derived from the action of SL(2,Z) on primitive n-squared origamis in the stratum H(2), showing most are non-planar by identifying K_{3,3} minors.

Contribution

It demonstrates that, except for two specific cases, the graphs associated with these origamis are non-planar, providing explicit minors to prove non-planarity.

Findings

Most graphs are non-planar, except for n=3 and one n=5 orbit.

Explicit K_{3,3} minors are identified in the graphs.

The non-planarity is established through graph minor analysis.

Abstract

We consider the $\text{SL}(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis in the stratum $\mathcal{H}(2)$. In particular, we consider the 4-valent graphs obtained from the action of $\text{SL}(2,\mathbb{Z})$ with respect to a generating set of size two. We prove that, apart from the orbit for $n = 3$ and one of the orbits for $n = 5$, all of the obtained graphs are non-planar. Specifically, in each of the graphs we exhibit a $K_{3,3}$ minor, where $K_{3,3}$ is the complete bipartite graph on two sets of three vertices.