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arXiv · 1803.05205

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

Moritz Firsching
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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.

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Moritz Firsching. 2018-03-14. The complete enumeration of 4-polytopes and 3-spheres with nine vertices. https://arxiv.org/abs/1803.05205

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