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 particular, we completely enumerate 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 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 are non-polytopal.

中文翻译：

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4-polytopes 和 3-spheres 九个顶点的完整枚举

我们描述了一种枚举多胞体的算法。然后实施该算法以给出具有 9 个顶点的 3 维组合球体的完整分类，并确定这些球体的多面性。特别是，我们完全枚举了所有具有 9 个顶点的 4 维多胞体的组合类型。结果表明，所有这些组合类型都是有理的：它们可以用有理坐标来实现。我们在 9 个顶点上发现了 316014 个组合球体。其中，274148 个可以实现为四维多面体的边界复合体，其余 41866 个是非多面体。