  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.
