Showing non-realizability of spheres by distilling a tree
| Autor: | Pfeifle, Julián |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. GAPCOMB - Geometric, Algebraic and Probabilistic Combinatorics |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Matemàtiques i estadística::Investigació operativa::Programació matemàtica [Àrees temàtiques de la UPC]
52 Convex and discrete geometry::52B Polytopes and polyhedra [Classificació AMS] Integer programming Programació en nombres enters 90 Operations research mathematical programming::90C Mathematical programming [Classificació AMS] |
| Popis: | In [Zhe20a], Hailun Zheng constructs a combinatorial 3-sphere on 16 vertices whose graph is the complete 4-partite graph K4;4;4;4. Such a sphere seems unlikely to be realizable as the boundary complex of a 4-dimensional polytope, but all known techniques for proving this fail because there are just too many possibilities for the 16 4 = 64 coordinates of its vertices. Known results [PPS12] on polytopal realizability of graphs also do not cover multipartite graphs. In this paper, we level up the old idea of Grassmann{Pl ucker relations, and assemble them using integer programming into a new and more powerful structure, called positive Grassmann{Pl ucker trees, that proves the non-realizability of this example and many other previously inaccessible families of simplicial spheres. See [Pfe20] for the full version |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|