Hyperbolic punctured spheres without arithmetic systole maximizers

We find bounds for the length of the systole – the shortest essential, non-peripheral closed curve – for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for $n=7,10,11$, arithmetic surfaces do not achieve the maximal systole length.