A multicurve${\mathcal{C}}$ in a closed orientable surface Sg of genus g is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. A left-handed Dehn twist$t_{\mathcal{C}}$about${\mathcal{C}}$ is the product of left-handed Dehn twists about the individual curves in ${\mathcal{C}}$. In this paper, we derive necessary and sufficient conditions for the existence of a root of $t_{\mathcal{C}}$ in the mapping class group Mod(Sg). Using these conditions, we obtain combinatorial data that correspond to roots, and use it to determine upper bounds on the degree of a root. As an application of our theory, we classify all such roots up to conjugacy in Mod(S4). Finally, we establish that no such root can lie in the level m congruence subgroup of Mod(Sg), for m ≥ 3.

Let Sg be a closed orientable surface of genus g ≥ 2 and C a simple closed nonseparating curve in F. Let tC denote a left-handed Dehn twist about C. A fractional power of tC of exponent ℓ//n is an h ∈ Mod(Sg) such that hn = tCℓ. Unlike a root of a tC, a fractional power h can exchange the sides of C. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if gcd(ℓ,n) = 1, then h will be isotopic to the ℓth power of an nth root of tC and that n ≤ 2g+1. In general, we show that n ≤ 4g, and that side-preserving fractional powers of exponents 2g//2g+2 and 2g//4g always exist. For a side-exchanging fractional power of exponent ℓ//2n, we show that 2n ≥ 2g+2, and that side-exchanging fractional powers of exponent 2g+2//4g+2 and 4g+1//4g+2 always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on S5.

Let $C$ be a curve in a closed orientable surface $F$ of genus $g\geq 2$ that separates $F$ into subsurfaces $\widetilde {{F}_{i} } $ of genera ${g}_{i} $, for $i= 1, 2$. We study the set of roots in $\mathrm{Mod} (F)$ of the Dehn twist ${t}_{C} $ about $C$. All roots arise from pairs of ${C}_{{n}_{i} } $-actions on the $\widetilde {{F}_{i} } $, where $n= \mathrm{lcm} ({n}_{1} , {n}_{2} )$ 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 $2{g}^{2} + 2g$, while $n\leq 4{g}^{2} + 2g$. We also give some additional applications.