Chainlink Polytopes and Ehrhart-Equivalence

TL;DR

This paper introduces chainlink polytopes derived from circular fence posets, enabling the construction of infinite non-isomorphic polytopes with identical Ehrhart quasi-polynomials, and proves a conjecture on unimodality.

Contribution

It defines chainlink polytopes and posets, revealing their symmetry properties and providing a method to generate Ehrhart-equivalent but non-isomorphic polytopes, and proves a conjecture on unimodality.

Findings

Constructed infinite families of Ehrhart-equivalent polytopes

Identified symmetry properties in circular fence and chainlink posets

Proved a conjecture on the unimodality of circular rank polynomials

Abstract

We introduce a class of polytopes that we call chainlink polytopes and which allow us to construct infinite families of pairs of non isomorphic rational polytopes with the same Ehrhart quasi-polynomial. Our construction is based on circular fence posets, which admit a non-obvious and non-trivial symmetry in their rank sequences that turns out to be reflected in the polytope level. We introduce the related class of chainlink posets and show that they exhibit the same symmetry properties. We further prove an outstanding conjecture on the unimodality of circular rank polynomials.