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

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

2008-01-16

arxiv.org/abs/0801.2582v1

Mathematics

Metric Geometry

14 pages

Scientific paper

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.

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