Polyhedral Embeddings of Triangular Regular Maps of Genus g, 1 < g < 15, and Neighborly Spatial Polyhedra

This article provides a survey about polyhedral embeddings of triangular regular maps of genus g, $2 \leq g \leq 14$, and of neighborly spatial polyhedra. An old conjecture of Gr\"unbaum 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 M\"obius torus. A numerical optimization approach of 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, $2 \leq g \leq 14$. The effectiveness of the new method has led to ten additional new polyhedral embeddings of triangular regular maps and their duals. There does exist symmetrical polyhedral embeddings of all triangular regular maps with genus g, $2 \leq g \leq 14$, except a single undecided case of genus 13.