Fullerene-like spheres with faces of negative curvature

Abstract

Given R\subset N, an (R,k)$-sphere is a k-regular map on the sphere whose faces have gonalities i\in R. The most interesting/useful are (geometric) fullerenes, i.e., (\{5,6\},3)$-spheres. Call \kappa_i=1 + \frac{i}{k} - \frac{i}{2} the curvature of i-gonal faces. (R,k)-spheres admitting \kappa_i<0 are much harder to study. We consider the symmetries and construction for three new instances of such spheres: ({a,b},k)-spheres with p_b\le 3 (they are listed), icosahedrites (i.e., ({3,4},5)$-spheres) and, for any c\in N, fullerene c-disks, i.e., ({5,6,c},3)-spheres with p_c=1.