Non-Realizable Minimal Vertex Triangulations of Surfaces: Showing Non-Realizability using Oriented Matroids and Satisfiability Solvers

TL;DR

This paper demonstrates the non-realizability of certain minimal vertex triangulations of surfaces in three-dimensional space by translating the problem into satisfiability and oriented matroids, providing new non-realizable examples and a general method.

Contribution

It introduces a novel approach using oriented matroids and satisfiability solvers to prove non-realizability of minimal triangulations of surfaces, including an infinite family of such examples.

Findings

No minimal vertex triangulation of genus 6 surfaces admits a polyhedral embedding in R^3.

Existence of minimal vertex triangulations of genus 5 surfaces that are non-realizable.

Development of a method applicable to other geometric realizability problems.

Abstract

We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R^3. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. We construct a new infinite family of non-realizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g. for face lattices of polytopes.