I am reading Farb and Margalit's A primer on Mapping Class groups which can be found in the following link: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf.
My question is regarding the relationship between the mapping class group of a $4$-punctured sphere and that of the torus, as described in section 2.2.5. On page 55 they discuss a construction of a $(p,q)$-curve on the torus in order to obtain a similar construction for a $(p,q)$-curve on a $4$-punctured sphere:
"Let $\alpha$ and $\beta$ be two simple closed curves in $T^2$ that intersect each other in one point. We identify $\alpha$ with $(1, 0) \in Z^2$ and $\beta$ with $(0, 1) ∈ Z^2$. Let $(p, q)$ be a primitive element of $Z^2$. A simple closed curve $\gamma$ in $T^2$ is a $(p, q)$- curve if we have $(i\hat (γ, β),i \hat(γ,α)) = ±(p, q)$. To construct the $(p, q)$-curve, we start by taking $p$ parallel copies of $α$, and we modify this collection by a $2π/q$ twist along $β$."
I cannot figure out what they mean by the last sentence . What does it mean to twist the $\alpha$-curves along $\beta$? Can you possibly help me understand this by pictures?
I agree with you that this language is a bit confusing. I had a discussion about this with Dan Margalit when I read that book.
Here is a local picture in a neighborhood of $\beta$.
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The horizontal arc is $\beta$ and the vertical arcs are the copies of $\alpha$. If we cut $\alpha$ at each intersection point with $\beta$ and then connect the $i-th$ strands of $\alpha$ on the bottom to the $(i+1)-st$ strand of alpha at the top (cyclically), we will get a connected curve. Try it on the torus for $p=2$ and $q=3$.