data set determines $t$ upto equivalence.

Nonseparating multicurves {#sec:nonsepmulcurves}
=========================

Recall that a multicurve ${\operatorname{{\mathcal{C}}}}$ is said to be nonseparating if it does not contain any pseudo-nonseparating submulticurves. In this section, we establish that a root of $t_{{\operatorname{{\mathcal{C}}}}}$ corresponds to a special kind of permuting action on the connected surface $S_g({\operatorname{{\mathcal{C}}}})$.

\[defn:orbit\_equiv\_sep\] Let $t_i$ be a permuting $(n_i,r_i,k_i)$-action on $S_{g_i}$ for $i =1,2$. Two orbits ${\operatorname{{\mathbb O}}}_i \in {\operatorname{{\mathbb O}}}(t_i)$ are said to be *equivalent* (in symbols, ${\operatorname{{\mathbb O}}}_1 \sim {\operatorname{{\mathbb O}}}_2$) if

(i) $|{\operatorname{{\mathbb O}}}_1| = |{\operatorname{{\mathbb O}}}_2|$, and

(ii) if $|{\operatorname{{\mathbb O}}}_1| < n := {\operatorname{lcm}}(n_1,n_2)$, then we further require that $$\theta_{{\operatorname{{\mathbb O}}}_1}(t_1) + \theta_{{\operatorname{{\mathbb O}}}_2}(t_2) \equiv 2\pi/n \pmod{2\pi}.$$

In this section, we will only need the case when $t_1 = t_2$, but we will need the general case in Section \[sec:sepmulcurves\].

\[def:nesnl2m\] Let ${\operatorname{{\mathcal{C}}}}$ be a nonseparating multicurve in $S_g$. A permuting $(n,2r,2k)$-action $t$ on $S_g({\operatorname{{\mathcal{C}}}})$ is said to be *nonseparating with respect to ${\operatorname{{\mathcal{C}}}}$* if there exists a $(r,k)$-partition ${\operatorname{{\mathbb P}}}_{r,k}({\operatorname{{\mathcal{C}}}})$ of ${\operatorname{{\mathcal{C}}}}$ such that

(i) there exists $r$ mutually disjoint pairs $\{P_i,P_i'\}$ of distinguished fixed points in ${\operatorname{{\mathbb P}}}(t)$ such that $\theta_{P_i}(t) + \theta_{P_i'}(t) = 2\pi/n$ modulo $2\pi$, for $1 \leq i \leq r$, and

