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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 $\phi(a)=C$.

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    $\begingroup$ 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. $\endgroup$ Nov 26, 2014 at 22:49

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Your second question is hard to understand. I assume you mean that the curve is simple, in which case, if the curve is non separating, it can be sent to one of the Humphries generators. If the curve is separating, you can still add all the Humphries generators to it and get a generating set, so what exactly do you mean?

As for the first question, you should explain what sort of algorithm you want (obviously, if you just keep writing words in, for example, the Humphries generators, you will hit your Dehn twist eventually, but this might not be what you seek).

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