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Neighborly 2-Manifolds with 12 Vertices

✍ Scribed by Amos Altshuler; Jürgen Bokowski; Peter Schuchert

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 516 KB
Volume: 75
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1996.0069

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✦ Synopsis

We provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space. Whereas the question of embeddability remains open in its general form, we obtain several properties of the embedding (polyhedral realization) under the assumption that it does exist:

The order of the geometrical automorphism group of any polyhedral realization would not exceed 2.
The polyhedral realization would not be obtainable via the Schlegel diagram of any 4-polytope; moreover, none of our orientable neighborly 2-manifolds with 12 vertices can be found within of the 2-skeleton of any 4-polytope.
The polyhedral realization would not allow a tetrahedral subdivision without inserting new vertices.

By using a weaker version of the manifold property, we obtain neighborly polyhedra with 2n vertices for every n 3.

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