Topological prismatoids and small simplicial spheres of large diameter
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad de Cantabria (UC) |
| Repositorio: | UCrea Repositorio Abierto de la Universidad de Cantabria |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.unican.es:10902/27811 |
| Acceso en línea: | https://hdl.handle.net/10902/27811 |
| Access Level: | acceso abierto |
| Palabra clave: | Simplicial complex Simplicial sphere Combinatorial diameter Hirsch conjecture |
| Sumario: | 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. |
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