Decomposition of a separating curve

Question

Consider a genus 2 surface. Its fundamental group can be expressed by $$ \pi_1(S_2)=\langle\alpha_1,\beta_1,\alpha_2,\beta_2\:||[\alpha_1,\beta_1][\alpha_2,\beta_2] \rangle. $$ I would like to know how to express the curve $\gamma$ as product of those generators when it is the separating curve that "separates one torus form the other": with "curve $\gamma$" I mean the pink one in this image.

Answer 1

A genus 2 surface can be visualized in terms of an octagon with boundaries identified as in this picture (this is a generalization of the usual way of seeing a torus as $\mathbb{R}^2/\mathbb{Z}^2$: analogously, a genus $n$ surface can be viewed as a $4n$-gon with appropriate boundary identifications).

The curve in question becomes the pink curve in this image, thus leading to (using your notation):

$$\gamma=[\alpha_1,\beta_1]=[\alpha_2,\beta_2]^{-1}$$

Edit: images taken from How to construct a genus 2 surface from 8-gon?.