Polyhedral Embeddings of Triangular Regular Maps of Genus g, 2 ⩽ g ⩽ 14, and Neighborly Spatial Polyhedra

This article provides a survey of polyhedral embeddings of triangular regular maps of genus g, 2⩽g⩽14, and of neighborly spatial polyhedra. An old conjecture of Grünbaum from 1967, although disproved in 2000, lies behind this investigation. We discuss all duals of these polyhedra as well, whereby we accept, e.g., the Szilassi torus with its non-convex faces to be a dual of the Möbius torus. A numerical optimization approach by the second author for finding such embeddings was first applied to finding (unsuccessfully) a dual polyhedron of one of the 59 closed oriented surfaces with the complete graph of 12 vertices as their edge graph. The same method has been successfully applied for finding polyhedral embeddings of triangular regular maps of genus g, 2⩽g⩽14. The effectiveness of the new method has led to ten additional new polyhedral embeddings of triangular regular maps and their duals. There do exist symmetrical polyhedral embeddings of all triangular regular maps with genus g, 2⩽g⩽14, except in a single undecided case of genus 13. Among these results, there are three new Leonardo polyhedra, each with 156 vertices, 546 edges, and 364 triangular faces, based on the Hurwitz triplet of genus 14 with Conder notation R14.1, R14.2, and R14.3.