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I am trying to figure out how to obtain generators for the mapping class group of a $4$-punctured sphere and I ran across this discussion in Farb and Margalit's A primer on Mapping Class Groups, which can be found here: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf.

In section 2.2.5 they talk about the relationship between the mapping class group of the $4$-punctured sphere and that of the torus. On page 55, they claim that if you take two curves $\alpha$ and $\beta$ that intersect at one point on $T^2$, then under the hyperelliptic involution they will descend to two closed curves $\bar{\alpha}$ and $\bar{\beta}$ on $S^2$. This is obviously incorrect for the standard meridian and longitude (${*}×S^1$ and $S^1×{*}$) on $T^2$, because (at least) one of them descends to an arc on $S^2$.

Can someone clarify the discussion in that section? I also do not entirely understand their construction of the $(p,q)$-curves which I posted on another question (mapping class group of a 4-punctured sphere). In any case, I would greatly appreciate the construction of generators (in terms of Dehn twists, I know there is a relationship with braid groups as well: $MCG(D_n,\partial D_n)\cong B_n$ , but I am not sure how half twists generators of of braid groups correspond to full Dehn twists) for the mapping class group of a $4$-punctured sphere even it is different from Farb and Margalit's.

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Perhaps this may clear things up: You don't have to choose "the standard" meridian and longitude, you can instead choose a meridian and longitude that do not pass through any of the four fixed points of the hyperelliptic involution.

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