Neighborly 2-manifolds with 12 vertices
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.
scientific article; zbMATH DE number 909619