$\gamma_j$ be the generators going around $q_j,1\leq j\leq s$. Let $x_p$ and $y_p,1\leq p\leq g_0$ be the standard generators of the “surface part” of ${\operatorname{{\mathcal O}}}$, chosen to give the following presentation of $\pi_1^{orb}({\operatorname{{\mathcal O}}})$: $$\begin{gathered}
\pi_1^{orb}({\operatorname{{\mathcal O}}}) = \langle \alpha_1,\alpha_2,\ldots, \alpha_r, \gamma_1,\gamma_2,\ldots, \gamma_s,x_1,y_1,x_2,y_2,\ldots x_{g_0},y_{g_0} \lvert \\ \alpha_1^n = \alpha_2^n = \ldots = \gamma_1^{n_1} = \gamma_2^{n_2} = \ldots = \gamma_s^{n_s} = 1, \alpha_1\ldots \alpha_r\gamma_1\ldots \gamma_s = \prod_{p=1}^{g_0}[x_p,y_p]\rangle\end{gathered}$$ From orbifold covering space theory [@T1], we have the following exact sequence $$1 \rightarrow \pi_1(S_g) \rightarrow \pi_1^{orb}({\operatorname{{\mathcal O}}}) \xrightarrow{\rho} C_n\rightarrow 1,$$ where $C_n = \langle t \rangle$. The homomorphism $\rho$ is obtained by lifting path representatives of elements of $\pi_1^{orb}({\operatorname{{\mathcal O}}})$. Since these do not pass through the cone points, the lifts are uniquely determined.

For $1\leq i\leq s$, the preimage of $q_i$ consists of $n/n_i$ points cyclically permuted by $t$. As in Notation \[defn:orbit\_distribution\], the rotation angle at each point is of the form $2\pi c_i^{-1}/n_i$ where $c_i$ is a residue class modulo $n_i$ and $gcd(c_i,n_i) = 1$. Lifting the $\gamma_i$, we have that $\rho(\gamma_i) = h^{(n/n_i)c_i}$. Similarly, lifting the $\alpha_i$ gives $\rho(\alpha_i)=t^{a_i}$ where $gcd(a_i,n)=1$. Finally, we have $$\rho(\prod_{p=1}^{g_0}[x_p,y_p]) = 1,$$ since $C_n$ is abelian, so $$1 = \rho(\alpha_1\ldots \alpha_r\gamma_1\ldots \gamma_s) = 