the centres of these pairs of disks by $Q_{i,j}$ and $Q_{i,j}'$, and the orbits of these centres by ${\operatorname{{\mathbb O}}}_j$ and ${\operatorname{{\mathbb O}}}_j'$. Thus $t$ is a permuting $(n,2r,2k)$-action with ${\operatorname{{\mathbb P}}}(t) = \{P_1,P_1',\ldots,P_r,P_r'\}$ and ${\operatorname{{\mathbb O}}}(t) = \{{\operatorname{{\mathbb O}}}_1,{\operatorname{{\mathbb O}}}_1',\ldots,{\operatorname{{\mathbb O}}}_k,{\operatorname{{\mathbb O}}}_k'\}$.

It remains to show that ${\operatorname{{\mathbb O}}}_i \sim {\operatorname{{\mathbb O}}}_i'$ for each $i$. By construction, $|{\operatorname{{\mathbb O}}}_i|=|{\operatorname{{\mathbb O}}}_i'| = m_i$, and if $m_i =n$, then ${\operatorname{{\mathbb O}}}_i\sim {\operatorname{{\mathbb O}}}_i'$ holds trivially. If not, then we need to show that the angle compatibility condition from Definition \[def:orbit\_equiv\] holds. Write ${\operatorname{{\mathbb O}}}_i= \{Q_{i,1},Q_{i,2},\ldots, Q_{i,m_i}\}$, ${\operatorname{{\mathbb O}}}_i' = \{Q_{i,1}',Q_{i,2}',\ldots, Q_{i,m_i}'\}$ and note that $h^{m_i}$ is an $(n/m_i)^{\text{th}}$ root of $t_{{\operatorname{{\mathcal{C}}}}}$ such that $h^{m_i}(c_{i,1}) = c_{i,1}$. Hence, $$\theta_{Q_{i,1}}(t^{m_i}) + \theta_{Q_{i,1}'}(t^{m_i}) = 2\pi/(n/m_i) \pmod{2\pi},$$ which implies that $$m_i\theta_{{\operatorname{{\mathbb O}}}_i}(t) + m_i\theta_{{\operatorname{{\mathbb O}}}_i'}(t) \equiv 2\pi/(n/m_i) \pmod{2\pi}.$$ Since $t^n$ is a restriction of a single Dehn twist, it follows that $$\theta_{{\operatorname{{\mathbb O}}}_i}(t) + \theta_{{\operatorname{{\mathbb O}}}_i'}(t) \equiv 2\pi/n \pmod{2\pi}.$$ Hence ${\operatorname{{\mathbb O}}}_i \sim {\operatorname{{\mathbb O}}}_i'$, and we obtain a permuting $(n,2r,2k)$-action $t$ on $S_g({\operatorname{{\mathcal{C}}}})$ that is nonseparating with respect to ${\operatorname{{\mathcal{C}}}}$.

Now suppose $h_1, h_2\in {\operatorname{Mod}}(S_g)$ are two roots of $t_{{\operatorname{{\mathcal{C}}}}}$ that are conjugate in ${\operatorname{Mod}}(S_g)$ via $\Phi \in {\operatorname{Mod}}(S_g)$, and let $t_s$ denote the 