$\mathcal{C}$ and takes $N$ to $N$.

By the Nielsen-Kerckhoff theorem [@K1], $\hat{t} := h\vert_{\widehat{S_g({\operatorname{{\mathcal{C}}}})}}$ is isotopic to a homeomorphism whose $n^{th}$ power is $\text{id}_{\widehat{S_g({\operatorname{{\mathcal{C}}}})}}$. So we may change $h$ by isotopy so that $\hat{t}^n = \text{id}_{\widehat{S_g({\operatorname{{\mathcal{C}}}})}}$. We fill in the $2m$ boundary circles of $\widehat{S_g({\operatorname{{\mathcal{C}}}})}$ with disks and extend $\hat{t}$ to a homeomorphism $t$ on $S_g({\operatorname{{\mathcal{C}}}})$ by coning. Thus $t$ defines a effective $C_n$-action on $S_g({\operatorname{{\mathcal{C}}}})$, where $C_n=\langle t\;\vert\;t^n=1\rangle$.

The $C_n$-action $t$ fixes the centers $P_i$ and $P_i'$ of the $2r$ disks $D_i$ and $D_i'$, $1 \leq i \leq r$, of $\overline{S_g\setminus \widehat{S_g({\operatorname{{\mathcal{C}}}})}}$ whose boundaries are the components of $\partial N$ which are fixed by $t$. The orientation of $S_g$ determines one for $S_g({\operatorname{{\mathcal{C}}}})$, so we may speak of directed angles of rotation about the centres of these disks. Since $h^n = t_{{\operatorname{{\mathcal{C}}}}}$, it follows from [@MK1 Theorem 2.1] that $$\theta_{P_i}(t) + \theta_{P_i'}(t) = 2\pi/n \pmod{2\pi},$$ as illustrated in Figure \[fig:twist\] below.

$P_i$ at 75 80 $P_i'$ at 270 115 $\theta_{P_i}(t)$ at 85 140 $\theta_{P_i'}(t)$ at 295 52

The remaining disks occuring in $\overline{S_g \setminus \widehat{S_g({\operatorname{{\mathcal{C}}}})}}$ form $k$ pairs of orbits of sizes $m_1, m_2,\ldots, m_k$ where $S({\operatorname{{\mathcal{C}}}}_{r,k}(h)) = {\Big\{\!\big\{}
\newcommand\rrbrace{\big\}\!\Big\}}m_1,m_2,\ldots, m_k\rrbrace$. For $1 \leq j \leq k$, we denote 