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

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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 order to decide polytopality, we generate polytopes by adding suitable points to polytopes with less than 9 vertices and therefore realize as many as possible of the combinatorial spheres as polytopes. For the rest, we prove non-realizability with techniques from oriented matroid theory. This yields a complete enumeration of 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 316 014 combinatorial spheres on 9 vertices. Of those, 274 148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41 866 are non-polytopal.

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Acknowledgments

I am very grateful to Günter M. Ziegler for insightful discussions and suggestions. I would like to thank an anonymous referee for valuable comments.

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Authors and Affiliations

  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195, Berlin, Germany

    Moritz Firsching

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  1. Moritz Firsching
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Correspondence to Moritz Firsching.

Additional information

This research was supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics.’

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Firsching, M. The complete enumeration of 4-polytopes and 3-spheres with nine vertices. Isr. J. Math. 240, 417–441 (2020). https://doi.org/10.1007/s11856-020-2070-4

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  • DOI: https://doi.org/10.1007/s11856-020-2070-4

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