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Polytopal and nonpolytopal spheres an algorithmic approach

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Abstract

The convexity theory for oriented matroids, first developed by Las Vergnas [17], provides the framework for a new computational approach to the Steinitz problem [13]. We describe an algorithm which, for a given combinatorial (d − 2)-sphereS withn vertices, determines the setC d,n(S) of rankd oriented matroids withn points and face latticeS. SinceS is polytopal if and only if there is a realizableM εC d,n(S), this method together with the coordinatizability test for oriented matroids in [10] yields a decision procedure for the polytopality of a large class of spheres. As main new result we prove that there exist 431 combinatorial types of neighborly 5-polytopes with 10 vertices by establishing coordinates for 98 “doubted polytopes” in the classification of Altshuler [1]. We show that for alln ≧k + 5 ≧8 there exist simplicialk-spheres withn vertices which are non-polytopal due to the simple fact that they fail to be matroid spheres. On the other hand, we show that the 3-sphereM 9963 with 9 vertices in [2] is the smallest non-polytopal matroid sphere, and non-polytopal matroidk-spheres withn vertices exist for alln ≧k + 6 ≧ 9.

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Authors and Affiliations

  1. Technische Hochschule Darmstadt, Fachbereich Mathematik, AG 3, Schlossgartenstrasse 7, 6100, Darmstadt, FRG

    Jürgen Bokowski & Bernd Sturmfels

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  1. Jürgen Bokowski
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  2. Bernd Sturmfels
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Bokowski, J., Sturmfels, B. Polytopal and nonpolytopal spheres an algorithmic approach. Israel J. Math. 57, 257–271 (1987). https://doi.org/10.1007/BF02766213

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