Topological Prismatoids and Small Simplicial Spheres of Large Diameter

TL;DR

This paper introduces topological prismatoids as a combinatorial abstraction to construct small, high-diameter simplicial spheres, demonstrating the existence of non-Hirsch spheres smaller than previously known examples.

Contribution

It extends the strong $d$-step theorem to topological prismatoids and constructs smaller non-$d$-step spheres using metaheuristic methods.

Findings

Existence of 8-dimensional spheres with 18 vertices exceeding the Hirsch bound.

Construction of four non-$d$-step 4-dimensional topological prismatoids with 14 vertices.

These spheres are shellable but their polytopality remains unknown.

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.