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

Author: | Moritz Firsching |

Journal: | Israel Journal of Mathematics, 2020. |

Full text: | arXiv • DOI • journal • PDF |

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.

```
@Article{ FirschingComplete4polys2020,
author = "Moritz Firsching",
title = "The complete enumeration of 4-polytopes and 3-spheres with nine vertices",
year = "2020",
journal = "Israel Journal of Mathematics",
doi = "10.1007/s11856-020-2070-4"
}
```