Roots of Dehn twists about separating curves

Let $C$ be a curve in a closed orientable surface $F$ of genus $g\ge 2$ that separates $F$ into subsurfaces $\widetilde{F_i}$ of genera $g_i$, for $i = 1,2$\mathrm{.}Westudythesetofrootsin$\Mod(F)$oftheDehntwist$t_C$about$C$\mathrm{.}Allrootsarisefrompairsof$C_{n_i}$-actionsonthe$\widetilde{F_i}$, where $n=\text{lcm}(n_1,n_2)$ is the degree of the root, that satisfy a certain compatibility condition. The $C_{n_i}$ actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, allowing a classification of all roots of $t_C$ by compatible pairs of data sets. We use these data set pairs to classify all roots for $g=2$ and $g=3$. We show that there is always a root of degree at least $2g^2+2g$, while $n \leq 4g^2+2g$. We also give some additional applications.