Abstract

We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the “strong d-step Theorem” that allows to construct such large-diameter polytopes from “non-d-step” prismatoids still works at this combinatorial level. Then, using metaheuristic methods on the flip graph, we construct four combinatorially different non-d-step 4-dimensional topological prismatoids with 14 vertices. This implies the existence of 8-dimensional spheres with 18 vertices whose combinatorial diameter exceeds the Hirsch bound. These examples are smaller that the previously known examples by Mani and Walkup in 1980 (24 vertices, dimension 11). Our non-Hirsch spheres are shellable but we do not know whether they are realizable as polytopes.

摘要

我们介绍了拓扑棱柱体，这是第二作者最近引入的（几何）棱柱体的组合抽象，用于构建赫希猜想的反例。我们表明，允许从“非d步”棱柱体构造这种大直径多面体的“强d步定理”在这个组合水平上仍然有效。然后，在翻转图上使用元启发式方法，我们构造了四个组合不同的 non- d- 具有 14 个顶点的 4 维拓扑棱柱体。这意味着存在具有 18 个顶点的 8 维球体，其组合直径超过赫希界。这些示例比 Mani 和 Walkup 在 1980 年已知的示例（24 个顶点，维度 11）要小。我们的非 Hirsch 球体是可壳的，但我们不知道它们是否可以实现为多面体。