# Dehn twist about an arbitrary curve

I need to know if there is an algortihm to write down a Dehn twist about an arbitrary curve on an orientable surface $S$ , as product of a set of generators of $MCG(S)$.

Since we have the conjugacy relation: $\phi^{-1}D_a\phi=D_{\phi(a)}$ ($D$ being the Dehn twist and equality understood up to isotopy), it will also help to know if for an arbitrary curve $C$ we can find a generator $a$(from any set of generators) and a diffeomorphism $\phi$ such that $\phi(a)=C$.

• In principle I think the answer is yes -- look at the Lickorish paper on the finite-generation of the mapping class group. This isn't the kind of procedure you would want to actually do, though. Nov 26, 2014 at 22:49