Although initially introduced to decipher the absolute Galois group of the field of rational numbers, dessins d’enfants (or equivalently bipartite ribbon graphs) have become a topic of interest in the context of both physics (e.g., Seiberg-Witten curves, [S. K. Ashok et al., Commun. Number Theory Phys. 1, No. 2, 237–305 (2007; Zbl 1230.14050)]) and mathematics (e.g., quivers, [Y.-H. He, “Calabi-Yau varieties: from quiver representations to dessins d’enfants”, Preprint, arXiv:1611.09398]). The category of bipartite ribbon graphs is equivalent to the category of Belyi pairs \((X,\beta)\) where \(X\) is an algebraic curve defined over a number field and \(\beta\) is a map from \(X\) to the projective line which is ramified at most over three points. From the viewpoint of both physics and mathematics, the explicit computation of Belyi maps is indispensable. Explicit computations become more difficult as the genus of the curve \(X\) grows.
In the article under review, given a rational function \(\varphi\), the authors consider the correspondence : \(Q(\varphi) = \frac{1}{2} + \frac{\sqrt{1-\varphi}}{2}\) which is relevant in the context of Painlevé VI equations. It turns out that these functions give rise the Belyi maps on elliptic curves, as exemplified in the article.