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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.
Comment: v1: 9 pages, 3 figures. v2: 10 pages, 4 figures. Final version with added references, discussion of alternate generating set, and changes suggested by the referee. To appear in Bull. Lond. Math. Soc |
