I am currently skipping through Farb&Margalits book about mapping class groups and trying to figure out their notation in their proof of the 9g-9 theorem (10.7.2) (p. 288 (in pdf 305):
We have a pants decomposition $\{\gamma_1, \dots, \gamma_{3g-3}\}$ (meaning $\gamma_i$ are certain simple, disjoint paths) and paths $\{\beta_1, \dots, \beta_{3g-3}\}$ of a surface s.t. "$\beta_i \cap \gamma_j = \emptyset \iff i \neq j$" holds (or more exactly $\beta$ and $\gamma$ notate their homotopie-class).
What is $\alpha_i:=T_{\gamma_i}(\beta_i)$ supposed to be?
Following the proof these so defined $\alpha_i$ seem to be paths(/their classes) which intersect similarly with the $\gamma_j$ as $\beta_i$ would. Hence my guess would be these $\alpha_i$ would be a twist (of degree 2$\pi$) of $\beta_i$ along $\gamma_i$, but I am not sure about that and can't find that notation anywhere else in this book. It would at least behave properly to the rest of the proof.
Anyone has seen this notation before and can confirm me?
It is exactly what you said it is, they also use the same notation at the beginning of chapter 3