The complete enumeration of 4-polytopes and 3-spheres with nine vertices

TL;DR

This paper presents a comprehensive algorithmic classification of all 4-polytopes and 3-spheres with nine vertices, including their realizability and rational coordinate representations, significantly advancing combinatorial polytope enumeration.

Contribution

It introduces a new enumeration algorithm and provides the complete classification of 4-polytopes and 3-spheres with nine vertices, including polytopality decisions and rational realizations.

Findings

Identified 316014 combinatorial spheres with 9 vertices

Found 274148 are realizable as 4-polytopes

Discovered 41866 are non-polytopal

Abstract

We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 are non-polytopal.