to the main result. In this paper, we shall derive conditions for the existence of a root of $t_{{\operatorname{{\mathcal{C}}}}}$ when $|{\operatorname{{\mathcal{C}}}}| \geq 2$, and since there are no such multicurves in $S_1$ or $S_0$, we shall assume henceforth that $g \geq 2$.

In general, a root $h$ of $t_{{\operatorname{{\mathcal{C}}}}}$ may permute some curves in ${\operatorname{{\mathcal{C}}}}$, while preserving other curves. So we define an *$(r,k)$-permuting root* of $t_{{\operatorname{{\mathcal{C}}}}}$ to be one that induces a partition of ${\operatorname{{\mathcal{C}}}}$ into $r$ singletons and $k$ other subsets of size greater than one. The theory for $(r,0)$-permuting roots, as we will see, can be obtained by generalizing the theories developed in [@MK1] and [@KR1], which involved the analysis of the fixed point data of finite cyclic actions.

The theory that we intend to develop for $(r,k)$-permuting roots when $k>0$ can be motivated by the following example. Consider the multicurve ${\operatorname{{\mathcal{C}}}}$ in $S_5$ shown in Figure \[fig:perm\_s5\].

$2\pi/5$ at 30 200 ![Nonseparating multicurve of size 5 in $S_5$[]{data-label="fig:perm_s5"}](Perm_S5 "fig:"){width="35"}

It is apparent that the rotation of $S_5$ by $2\pi/5$ composed with $t_c$ for some fixed $c\in {\operatorname{{\mathcal{C}}}}$ is a 5$^{th}$ root of $t_{{\operatorname{{\mathcal{C}}}}}$ in ${\operatorname{Mod}}(S_5)$. This is a simple example of a $(0,1)$-permuting root, which is 