This paper by Gulotta contains the dessins algorithm (of course the stringers like to talk about dimers and AdS/CFT). In the previous example, there were two columns of the $2 \times 6$ array that formed the $2 \times 2$ matrix shown here.
This diagram illustrates a typical replacement of four paths on a torus, of winding number $\pm 1$, by two paths with opposite winding numbers, defining the columns of the matrix. For general $2 \times m$ arrays one can replace single intersection points, four intersection points (as drawn here), and so on. Since the twisting of curves must avoid further intersections, there is a very limited number of moves and it is fairly easy to construct a diagram from a given array. Torus dessins (ribbon graphs) from secondary polytopes! Grothendieck would love it.
6 years ago