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Chainlink Polytopes and Ehrhart-Equivalence

Ezgi Kantarcı Oğuz, Cem Yalım Özel, Mohan Ravichandran

TL;DR

The paper introduces chainlink polytopes $\mathrm{CL}(\bar{a},l)$ and their sections $\mathrm{CL}^t(\bar{a},l)$, motivated by fence and circular fence posets. It proves a striking symmetry: for $2l\le\min_i a_i$, the Ehrhart quasi-polynomials of complementary sections coincide, yielding Ehrhart-equivalence for many non-isomorphic polytopes, and it builds a matrix-analytic framework using oriented posets to establish rank-symmetry for chainlink posets and their circular-fence relatives. The authors then prove unimodality results for circular fence rank polynomials, identify exceptional cases, and show that stretching can induce multimodality, while also deriving structural properties (vertices, volumes, Minkowski sums) of chainlink polytopes. Their approach connects polyhedral geometry with poset theory via a transparent linear-algebraic method, providing new proofs of known symmetry results and new avenues for Ehrhart-theoretic investigations of polytope families. The work thereby advances understanding of symmetry, unimodality, and equivalence phenomena in a rich class of rational polytopes linked to combinatorial posets.

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.

Chainlink Polytopes and Ehrhart-Equivalence

TL;DR

The paper introduces chainlink polytopes and their sections , motivated by fence and circular fence posets. It proves a striking symmetry: for , the Ehrhart quasi-polynomials of complementary sections coincide, yielding Ehrhart-equivalence for many non-isomorphic polytopes, and it builds a matrix-analytic framework using oriented posets to establish rank-symmetry for chainlink posets and their circular-fence relatives. The authors then prove unimodality results for circular fence rank polynomials, identify exceptional cases, and show that stretching can induce multimodality, while also deriving structural properties (vertices, volumes, Minkowski sums) of chainlink polytopes. Their approach connects polyhedral geometry with poset theory via a transparent linear-algebraic method, providing new proofs of known symmetry results and new avenues for Ehrhart-theoretic investigations of polytope families. The work thereby advances understanding of symmetry, unimodality, and equivalence phenomena in a rich class of rational polytopes linked to combinatorial posets.

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.
Paper Structure (13 sections, 23 theorems, 71 equations, 7 figures, 2 tables)

This paper contains 13 sections, 23 theorems, 71 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Chainlink polytopes
  4. Connections to other polytopes [Ezgi, Mohan]
  5. Polyhedral models for fence posets
  6. chainlink Posets and Rank Symmetry
  7. Matrix Formulation
  8. Recurrence Relations and Rank Symmetry
  9. Unimodality and Multimodality
  10. Unimodality
  11. Multimodality
  12. Properties of Chainlink polytopes
  13. Remarks and further work

Key Result

Theorem 1

Let $\bar{a} = (a_1, \ldots, a_s)$ be a composition of $n$, let $l$ be a non-negative integer such that $2l \leq \operatorname{min} \{a_i\}_{i \in [s]}$ and let $t$ be a positive integer. Then complementary sections of the chainlink polytope have the same Ehrhart quasipolynomial,

Figures (7)

  • Figure 1: The polytope $\mathrm{CL}(\bar{a}= (6, 4, 5), \, l = 2)$
  • Figure 2: The sections $CL^4(\bar{a}, l)$and $CL^{11}(\bar{a}, l)$ where $\bar{a} = (6, 4, 5)$ and $l = 2$.
  • Figure 3: The fence poset $F(2,1,1,2)$ (left) and two depictions of the circular fence poset $\bar{F}(2, 1, 1, 2)$(center and right). In the middle one, the two nodes marked $x_1$ are identified.
  • Figure 4: A chainlink poset: $C((4, 5), 2)$
  • Figure 5: The chainlink poset with $\bar{a}=(6,4,5)$ and $l=2$, the two top right nodes are connected to the two bottom left nodes.
  • ...and 2 more figures

Theorems & Definitions (45)

  • Theorem 1
  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4: main
  • Conjecture 2.5: main
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3: Stanley
  • ...and 35 more