Dessin D'enfant - Maps and Hypermaps

Maps and Hypermaps

A vertex in a dessin has a graph-theoretic degree, the number of incident edges, that equals its degree as a critical point of the Belyi function. In the example above, all white points have degree two; dessins with the property that each white point has two edges are known as clean, and their corresponding Belyi functions are called pure. When this happens, one can describe the dessin by a simpler embedded graph, one that has only the black points as its vertices and that has an edge for each white point with endpoints at the white point's two black neighbors. For instance, the dessin shown in the figure could be drawn more simply in this way as a pair of black points with an edge between them and a self-loop on one of the points. It is common to draw only the black points of a clean dessin and to leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map.

Thus, any embedding of a graph on a surface in which each face is a disk (that is, a topological map) gives rise to a dessin by treating the graph vertices as black points of a dessin, and placing white points at the midpoint of each embedded graph edge. If a map corresponds to a Belyi function ƒ, its dual map (the dessin formed from the preimages of the line segment ) corresponds to the multiplicative inverse 1/ƒ.

A dessin that is not clean can be transformed into a clean dessin on the same surface, by recoloring all of its points as black and adding new white points on each of its edges. The corresponding transformation of Belyi pairs is to replace a Belyi function β by the pure Belyi function γ = 4β(β − 1). One may calculate the critical points of γ directly from this formula: γ−1(0) = β−1(0) ∪ β−1(1), γ−1(∞) = β−1(∞), and γ−1(1) = β−1(1/2). Thus, γ−1(1) is the preimage under β of the midpoint of the line segment, and the edges of the dessin formed from γ subdivide the edges of the dessin formed from β.

Under the interpretation of a clean dessin as a map, an arbitrary dessin is a hypermap: that is, a drawing of a hypergraph in which the black points represent vertices and the white points represent hyperedges.