Polyhedral Embeddings and Immersions of Many Triangulated 2-Manifolds with Few Vertices

TL;DR

This paper improves a heuristic for embedding triangulated 2-manifolds, enabling realizations of non-orientable surfaces with few vertices, including symmetric cases, leading to new minimal and near-minimal surface embeddings.

Contribution

It extends a previous heuristic to non-orientable surfaces and symmetric realizations, allowing for the first time the realization of minimal or near-minimal non-orientable surfaces.

Findings

Realized non-orientable surfaces with minimal vertices

Achieved symmetric realizations of complex surfaces

First-time embeddings of certain projective planes and Klein bottles

Abstract

This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one or two handles and the Klein Bottle with one or two handles.