A flag vector of a 3-sphere that is not the flag vector of a 4-polytope

TL;DR

This paper provides the first example of a 3-sphere with a specific flag vector that cannot be realized as a convex 4-polytope, demonstrating its non-polytopality through computer-assisted proofs involving oriented matroids.

Contribution

It introduces a unique 3-sphere with a specific flag vector that is not realizable as a polytope, and provides computer-verified proofs of its non-polytopality.

Findings

The 3-sphere with parameters (12,40,40,12;120) is unique.

This sphere is not realizable as a convex 4-polytope.

The sphere cannot be embedded as a polytopal complex.

Abstract

We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters $(f_0,f_1,f_2,f_3;f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex 4-polytope. The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.