A Belyi function is a ramified cover $\beta : X \rightarrow \mathbb{P}^1\mathbb{C} $ unramified outside $\{ 0,1,\infty\}$.
Now if I colour the vertex 0 black, 1 white and look at the preimage $\beta^{-1}([0,1])$, I understand that we get a connected bipartite graph and the valency of the black and white vertices in the preimage are the multiplicities of the critical points, as on a neighbourhood of 0 and 1, $\beta$ looks like $z \mapsto z^k$.
What I fail to see however, is why the degree of a pole is equal to the number of edges incident to a face.
Can someone help me out with why this is? Thank you!
Draw a triangle $T$ in the Riemann sphere, with vertices $0$, $1$ and $\infty$, and colour the edges red, blue and green say. Now look at the inverse image $T'$ of $T$. This is a triangulation of $X$ and we colour its edges according to their image in $T$. Let the $0$-$1$ edge be red. At a pole of $\beta$, $k$ blue and $k$ green edges emerge, where $k$ is the order of the pole. So $2k$ triangles of $T$ meet at that point, and they combine to form a face of $\beta^{-1}(T)$ with $2k$ edges. (The faces of $\beta^{-1}(T)$ have an even number of edges since it's a bipartite graph).