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