Constructions for 4-Polytopes and the Cone of Flag Vectors

TL;DR

This paper introduces a new construction method for 4-polytopes that generalizes stacking, enabling the creation of specific 4-polytopes with particular properties and revealing insights into the structure of their flag vectors.

Contribution

It presents a novel construction technique for 4-polytopes that extends stacking, demonstrating the inclusion of a specific ray in the convex hull of flag vectors.

Findings

Constructed 4-polytopes with g_2=0 for n >= 13 vertices

Showed the ray l_1 is in the convex hull of flag vectors

Presented examples with 9, 10, and 11 vertices

Abstract

We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g_2=0 on any number of n >= 13 vertices. In particular, this implies that the ray l_1, described by Bayer (1987), is fully contained in the convex hull of all flag vectors of 4-polytopes. Especially interesting examples on 9, 10 and 11 vertices are presented.