Gap between the number of facets of the two poset polytopes

TL;DR

This paper investigates the difference in the number of facets between order and chain polytopes of posets, introducing crossing numbers to establish bounds and classify posets with minimal gaps.

Contribution

It introduces crossing numbers to quantify the facet gap and classifies posets with gaps of zero and one, advancing understanding of poset polytope structures.

Findings

Classified posets with zero facet gap.

Established bounds on the facet gap using crossing numbers.

Identified posets with a gap of exactly one.

Abstract

We study the difference between the number of facets of the order polytope and the chain polytope of a poset. Hibi and Li classified posets where the gap is exactly zero. We describe the bounds on this gap using the new notion of crossing numbers, and then use this result to classify the posets where the gap is exactly one.