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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