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I am looking for a suitable set of Dehn-twists generators for the mapping class group of a curve of genus $g$ with $n$ marked points (i.e. the mapping class group of $\mathcal M_{g,n}$).

For $\mathcal M_g$ one can use $2g+1$ Dehn twists, and also choose them as non-separating loops (i.e. they do not split the surface in two disconnected components). This is what I need.

For $\mathcal M_{g,n}$ I would imagine that one should add $n-1$ ($n$?) extra Dehn twists around pairs of marked points to generate the elementary braids. Since I am not well read in the topic I am looking for information/references.

Ideally I would like that there exist a suitable number ($2g+n$ more or less) of Dehn twists, each of which have the following generalized nonseparating property;

either they do not separate the pointed curve or they separate but in each component there is at least one puncture.

Thanks for your help!

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The answer to your question can be found on page 114 of Farb-Margalit's Primer on Mapping Class Groups. See Figure 4.10. All the Dehn twists they list are actually nonseparating curves.

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