A textit{multicurve} $C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{C}$ about $C$ is the product of the Dehn twists about the individual curves. In t
his paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.
Let $S_g$ be a closed orientable surface of genus $g geq 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A textit{fractional power} of $t_C$ of textit{exponent} $fraction{ell}{n}$ is an $h i
n Mod(S_g)$ such that $h^n = t_C^{ell}$. Unlike a root of a $t_C$, 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 $gcd(ell,n) = 1$, then $h$ will be isotopic to the $ell^{th}$ power of an $n^{th}$ root of $t_C$ and that $n leq 2g+1$. In general, we show that $n leq 4g$, and that side-preserving fractional powers of exponents $fraction{2g}{2g+2}$ and $fraction{2g}{4g}$ always exist. For a side-exchanging fractional power of exponent $fraction{ell}{2n}$, we show that $2n geq 2g+2$, and that side-exchanging fractional powers of exponent $fraction{2g+2}{4g+2}$ and $fraction{4g+1}{4g+2}$ always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on $S_5$.
