an example of a such a surface $S_{22}$ with a multicurve ${\operatorname{{\mathcal{C}}}}= {\operatorname{{\mathcal{C}}}}^{(2)}(2) \sqcup {\operatorname{{\mathcal{C}}}}^{(1)}(3)$ constructed as in Notation \[defn:pseudo\_multicurve\] from the surfaces $S_5, {\operatorname{{\mathbb S}}}_3(2)$, and ${\operatorname{{\mathbb S}}}_3(3)$.

${\operatorname{{\mathcal{C}}}}^{(2)}(2)$ at 70 95 ${\operatorname{{\mathcal{C}}}}^{(1)}(3)$ at 620 35

\[rem:root\_isotopy\] Let ${\operatorname{{\mathcal{C}}}}= \{c_1,c_2,\ldots, c_m\}$, and suppose that $h$ is a root of $t_{{\operatorname{{\mathcal{C}}}}}$ of degree $n$ in ${\operatorname{Mod}}(S_g)$. Then, we claim that, up to isotopy, $h({\operatorname{{\mathcal{C}}}}) = {\operatorname{{\mathcal{C}}}}$.

Suppose first that $h(c_i) \nsim c_1$ for all $i$. Then there exists a neighbourhood $N$ of $c_1$ such that $$t_{h(c_i)} \mid_N = \text{id}_N \text{ and } t_{c_i}\mid_N = \text{id}_N \quad\forall i\neq 1$$ However, since $h^n = t_{{\operatorname{{\mathcal{C}}}}}$, it follows that $$t_{{\operatorname{{\mathcal{C}}}}} = ht_{{\operatorname{{\mathcal{C}}}}}h^{-1} = ht_{c_1}h^{-1}ht_{c_2}h^{-1}\ldots ht_{c_m}h^{-1} = t_{h(c_1)}t_{h(c_2)}\ldots t_{h(c_m)}$$ Hence $$t_{c_1}\mid_N = t_{{\operatorname{{\mathcal{C}}}}}\mid_N = t_{h(c_1)}t_{h(c_2)}\ldots t_{h(c_m)}\mid_N = \text{id}_N$$ which is a contradiction.

So we assume without loss of generality that $h(c_1) \sim c_1$. Then we choose a neighbourhood $N'$ of $c_1$ disjoint from $c_i$ for $i\neq 1$, and choose an isotopy $\varphi_t$ such that $$\varphi_t\mid_{S_g\setminus N'} = \text{id}_{S_g\setminus N'}$$ and $\varphi_0 = h$ and $\varphi_1(c_1) = c_1$. Replacing $h$ by $\varphi_1$, we may assume that $h(c_1) = c_1$. Now note that $h(c_i) \nsim c_1$ for all $i\neq 1$, which allows us to proceed by 