Neighborly 2-manifolds with 12 vertices
type: Item
instance of: scholarly article
title: Neighborly 2-manifolds with 12 vertices
zbMATH Open document ID: 0855.52002
author: Amos Altshuler
author: Jürgen Bokowski
author: Peter Schuchert
published in: Journal of Combinatorial Theory. Series A
publication date: 1997-02-16T00:00:00Z
review text: It is a long-standing open problem whether there exists a piecewise flat embedding of a surface of genus 6 into 3-space with only 12 vertices, necessarily with complete edge graph. In this interesting paper a complete enumeration of all such abstract 12-vertex triangulations is given, and 59 items are found. The main problem remains open. However, it is shown that piecewise flat embeddings into 3-space cannot have symmetry groups of order greater than 2. Furthermore none of these triangulations can be contained in the 2-skeleton of any convex 4-polytope. In addition all possible tetrahedral decomposition of the `interiors' of these surfaces are found, i.e., all 12-vertex triangulations of the 3-dimensional handlebody of genus 6. It turns out that there are precisely 68 triangulated 3-manifolds with 12 vertices containing a triangulated surface of genus 6. All of them are non-polytopal 3-spheres.
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 05C30
zbMATH DE Number: 909619
zbMATH Keywords: neighborly 2-manifolds
zbMATH Keywords: tetrahedral subdivision
zbMATH Keywords: matroid polytope
zbMATH Keywords: flat embeddings
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: jcta.1996.0069
OpenAlex ID: W2076574221
DOI: 10.1006/JCTA.1996.0069
description: scientific article; zbMATH DE number 909619