Non-existence of polyhedral immersions of triangulated surfaces in $\mathbb{R}^3$

TL;DR

This paper introduces a method to prove the non-existence of certain polyhedral immersions of triangulated non-orientable surfaces in three-dimensional space, demonstrating that specific minimal triangulations cannot be realized as immersed polyhedral surfaces.

Contribution

It provides a novel approach for disproving the realizability of particular triangulations of non-orientable surfaces in 3D space, focusing on vertex-minimal, neighborly cases.

Findings

Neither of the two vertex-minimal, neighborly 9-vertex triangulations of the non-orientable surface of genus 5 are realizable as immersed polyhedral surfaces in R^3.

The method effectively disproves the existence of certain polyhedral immersions.

The results contribute to understanding the limitations of polyhedral realizations of non-orientable surfaces.

Abstract

We present and apply a method for disproving the existence of polyhedral immersions in $\mathbb{R}^3$ of certain triangulations on non-orientable surfaces. In particular, it is proved that neither of the two vertex-minimal, neighborly 9-vertex triangulations of the non-orientable surface of genus 5 are realizable as immersed polyhedral surfaces in $\mathbb{R}^3$.