## Classifying, realizing ...

### ... all combinatorial 3-spheres and 4-polytopes with up to 9 vertices

All combinatorial types are preceded by the flag f-vector $(f_0, f_1, f_2, f_3 ), f_{0,3}$.n=5 | n=6 | n=7 | n=8 | n=9 | 5<=n<=9 | |

combinatorial 3-spheres | 1 | 4 | 31 | 1336 | 316014 | tar.gz |

4-polytopes | 1 | 4 | 31 | 1294 | 274148 | tar.gz |

Each line contains a combinatorial sphere.

$(f_0, f_1, f_2, f_3 ), f_{0,3}$, [polytope|nonrealizable] type, additional data.

If the sphere is the boundary of a polytope, we provide rational coordinates of such a polytope.

Otherwise we provide a non-realizability certificate of a certain type. where "type" can be one of three values:

- this combinatorial type has a partial chirotope that contradicts Graßmann-Plücker: the partial chirotope and the GP-relation violated follows
- this combinatorial type has a partial chirotope which can't be completed consistently. Relevant GP-relations and the contradiction follow
- the completed chirotope admits a biquadratic final polynomial. We provide the completed chirotope and the linear program, which is infeasible

### ... and inscribing simplicial 3-spheres with up to 10 vertices

Numbering corresponds to the enumeration of Frank Lutz. This is a table with the number simplicial 3-spheres with n vertices. Click on the file to download a file with rational realizations. Most of the realizations are inscribed in the unit sphere, the second row states how many of the realizations are inscribed.n=5 | n=6 | n=7 | n=8 | n=9 | n=10 | |

d=4 | 1 | 2 | 5 | 37 | 1142 | 162004 |

inscribed | 1 | 2 | 5 | 37 | 1140 | 161978 |

### ... neighborly uniform oriented matroids

Numbering corresponds to the enumeration by Hiroyuki Miyata (宮田 洋行). Here is a table with the number of (simplicial) neighborly d-polytopes with n vertices for small d and n. Click on the number to download a file with rational realizations of these polytopes on the sphere. Some of the files are rather large.n=5 | n=6 | n=7 | n=8 | n=9 | n=10 | n=11 | n=12 | |

d=4 | 1 | 1 | 1 | 3 | 23 | 431 | 13935 | 556061 |

d=5 | 1 | 1 | 2 | 126 | 159375 | |||

d=6 | 1 | 1 | 1 | 37 | 42099 | |||

d=7 | 1 | 1 | 4 | 35993 |