I need help finding a nice way to define coordinates on a surface of genus two and construct its De Rham cohomology ring explicitly in terms of differential forms generating it.
What I have done: I know how to construct a surface of genus two using an octagon by gluing the edges. This way, we have a vertex, four loops ($a$, $b$, $c$, and $d$), and a disk with its boundary identified by $dcbab^{-1}a^{-1}d^{-1}c^{-1}$. For a simple torus, we only have two loops, and we could just parametrize each one of them by an angle which would also give us the coordinates on the torus. But here, we have four loops, but we only need two variables to define coordinates on the space. How do I fix this?
Note: I do not have algebraic topology background, so if you're using a fancy algebraic trick or theorem, please give a geometric explanation.