Euler characteristic of a paired-polygon surface cover ($\chi (M)=|T|(1-s+{\sum }_j{q}_j^{-1})$)

For a paired $2s$-gon with $r$ base vertex cycles of a paired polygon, there are one face, $s$ edges and $r$ vertices on the base, so $\chi (M_0)=1-s+r$. The full finite-group construction has $|T|$ faces, $s|T|$ edges and ${\sum }_j|T|/q_j$ vertices, giving the displayed ordinary Euler characteristic. If all lifted angles are $2\pi$, the orbifold Euler characteristic formula gives equivalently $\chi (M)=|T|{\chi }_{\mathrm{orb}}(M_0)$.