  Let $S_g$ denote the closed orientable surface of genus $g$. We construct
exponentially many mapping class group orbits of collections of $2g+1$ simple
closed curves on $S_g$ which pairwise intersect exactly once, extending a
result of the first author and further answering a question of
Malestein-Rivin-Theran. To distinguish such collections up to the action of the
mapping class group, we analyze their dual cube complexes in the sense of
Sageev. In particular, we show that for any even $k$ between $\lfloor g/2
\rfloor$ and $g$, there exists such collections whose dual cube complexes have
dimension $k$, and we prove a simplifying structural theorem for any cube
complex dual to a collection of curves on a surface pairwise intersecting at
most once.
