Complex parametrization of triangulations on oriented maps

Abstract

We consider here triangulations of oriented maps having a specified set S of vertices of degree different from 6 and some other vertices of degree 6. Such map can be described by specifying the relative positions between elements of S using Eisenstein integers. We first consider the case of 1 parameter, which corresponds to the Goldberg-Coxeter construction. Then we develop the general theory, the special case of positive curvature studied by Thurston and finally extend the theory to quadrangulations and some other cases. In the last section we expose application of parameterizations to the study of zigzags.