Let $S\_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S\_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (equivalently think of it as the mapping class group of the once-punctured surface of genus $g$).


Is every element of $Mod(S\_{g,1})$ a composition of right-handed Dehn twists?


Note that this is true for $S\_{g,0}$ as stated in page 124 of *A primer on Mapping Class Groups* by Farb and Margalit under the name of "a strange fact".


Edit: I am going to comment ThiKu's answer to avoid further confusion. 


In [A. Wand: Factorisation of Surface Diffeomorphisms](https://arxiv.org/pdf/0910.5691.pdf) and in [Baker, Etnyre and Van Horn-Morris: Cabling, Contact Structures and and Mapping Class Monoids](https://arxiv.org/pdf/1005.1978.pdf) the authors, independently, provide with examples of diffeomorphisms in $Veer(\Sigma\_{2,1},\partial \Sigma\_{2,1})$ which are not in $Dehn^+(\Sigma\_{2,1}, \partial \Sigma\_{2,1})$. That is, right-veering diffeomorphisms which are not a product of right-handed Dehn twists. However, the mapping class group in which these results hold is $Mod( \Sigma\_{2,1}, \partial \Sigma\_{2,1})$, that is, the mapping class group of automorphisms fixing the boundary and isotopies fixing the boundary as well. This, a priori, does not yield counter-examples to my question (unless it does together with some other result that I do not know).


Observe that for all $g \geq 1$ there is a central extension 


$$1 \to \mathbb{Z} \to Mod(\Sigma\_{g,1}, \partial \Sigma\_{g,1}) \to Mod( \Sigma\_{g,1}) \to 1 $$


which is not split in general.



Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?
Yes. As pointed out by Ian Agol, I actually answered my question.


I observed that the identity is a non-empty composition of right handed Dehn twists in $Mod(S\_{g,1})$. A priori this is not trivial. I was thinking about monodromies on Brieskorn-Pham singularities $(x^p+y^q)$ which are freely periodic and a composition of right-handed Dehn twists (by morsifying the singularity). One can easily see that this solves the problem in the cases that I was asking originally since all surfaces $S\_{g,1}$ appear as Milnor fibers of such singularities for all $g$.


EDIT: I removed the last part of the answer since it was not true in that generality and this answers completely what I wanted.


