The complete enumeration of 4-polytopes and 3-spheres with nine vertices
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THE COMPLETE ENUMERATION OF 4-POLYTOPES AND 3-SPHERES WITH NINE VERTICES
BY
Moritz Firsching∗ Institut f¨ ur Mathematik, Freie Universit¨ at Berlin Arnimallee 2, 14195 Berlin, Germany e-mail: [email protected]
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.
1. Introduction 1.1. Results. Having good examples (and counterexamples) is essential in discrete geometry. To this end, a substantial amount of work has been done on the classification of polytopes and combinatorial spheres; see Subsection 1.4. The ∗ This research was supported by the DFG Collaborative Research Center TRR
109, ‘Discretization in Geometry and Dynamics.’ Received April 9, 2018 and in revised form February 2, 2020
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M. FIRSCHING
Isr. J. Math.
classification of combinatorial 3-spheres and 4-polytopes for 7 vertices was done by Perles, [Gr¨ u67, Section 6.3] and for 8 vertices it was completed by Altshuler and Steinberg [AS85]. As a next step, we present new algorithmic techniques to obtain a complete classification of combinatorial 3-spheres with 9 vertices into polytopes and nonpolytopes. We obtain the following results: Theorem 1: There are precisely 316 014 combinatorial types of combinatorial 3-spheres with 9 vertices. Theorem 2: There are precisely 274 148 combinatorial types of 4-polytopes with 9 vertices. Therefore we have 41 866 non-polytopal combinatorial types of combinatorial 3-spheres with 9 vertices. By taking polar duals, we immediately also obtain a complete classification of 4-polytopes and 3-spheres with nine facets. We provide rational coordinates for all of the combinatorial types of 4-polytopes with 9 vertices. We call a polytope rational if it is combinatorially equivalent to a polytope with rational coordinates. Corollary 3: Every 4-polytope with up to 9 vertices is rational. Every 4-polytope with up to 9 facets is rational. Perles showed, using Gale diagrams, that all d-polytopes with at most d + 3 vertices are rational; see [Gr¨ u67, Chapter 6]. There is an example of Perles of an 8-polytope with 12 vertices, which is not rational [Gr¨ u67, Thm. 5.5.4, p. 94]. For d = 4, there are examples of non-rational polytopes with 34 [RGZ95, Corollary of Mai
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