arxiv: 2604.23848 · v1 · submitted 2026-04-26 · 🧮 math.SG · math.CO

Recognition: unknown

Contact flexibility and rigidity for toric Gorenstein prequantizations and Ehrhart theory of toric diagrams

Ant\'onio Rocha-Neves, Leonardo Macarini, Miguel Abreu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:41 UTC · model grok-4.3

classification 🧮 math.SG math.CO

keywords Gorenstein toric contact manifoldstoric diagramsEhrhart polynomialscontact Betti numbersrigidityflexibilityprequantizationscross-polytope

0 comments

The pith

The small cross-polytope is the unique toric diagram sharing the Ehrhart polynomial of the cross-polytope, making the primitive prequantization of P^1 × ⋯ × P^1 rigid among Gorenstein toric contact manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Gorenstein toric contact manifolds with zero first Chern class are classified by integral convex polytopes called toric diagrams, where the diagram's Ehrhart polynomial fixes the dimensions of the cylindrical contact homology in each degree. The paper demonstrates that flexibility is typical: the toric diagrams of prequantizations of monotone iterated P^1-bundles all share the same Ehrhart polynomial as the cross-polytope coming from the product of projective lines, and the authors give a unimodular classification of these diagrams. In contrast, the specific diagram they call the small cross-polytope is shown to be the only one with this Ehrhart polynomial in every dimension. This uniqueness proves that the contact Betti numbers determine the manifold completely in this rigid case. A parallel uniqueness result holds for the toric diagrams of primitive prequantizations of monotone P^1-bundles over projective space.

Core claim

Gorenstein toric contact manifolds are determined by toric diagrams whose Ehrhart polynomials encode the contact Betti numbers. While the family of toric diagrams arising from monotone iterated P^1-bundles share the Ehrhart polynomial of the cross-polytope, the small cross-polytope is the unique toric diagram with this polynomial in each dimension, so the primitive prequantization of P^1 × ⋯ × P^1 is rigidly determined by its contact Betti numbers. The same rigidity holds for the family of primitive prequantizations of monotone P^1-bundles over P^{n-1}.

What carries the argument

Toric diagram, the integral convex polytope that encodes a Gorenstein toric contact manifold and whose Ehrhart polynomial determines the contact Betti numbers.

If this is right

In each dimension the small cross-polytope is the only toric diagram with its Ehrhart polynomial.
Contact Betti numbers determine the Gorenstein toric contact manifold uniquely for the primitive prequantization of the product of projective lines.
The toric diagrams of monotone Bott manifolds are classified up to unimodular equivalence by their shared Ehrhart polynomial.
Rigidity holds for the family of primitive prequantizations of monotone P^1-bundles over P^{n-1}.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests that Ehrhart polynomials can serve as complete invariants for certain classes of toric diagrams in contact geometry.
Similar rigidity statements might be provable for other distinguished toric diagrams beyond the small cross-polytope.
The flexibility shown for the Bott manifold family indicates that many geometrically distinct prequantizations can share the same combinatorial contact invariants.

Load-bearing premise

Gorenstein toric contact manifolds with zero first Chern class are completely determined by their toric diagrams, and the Ehrhart polynomial of the diagram exactly determines the contact Betti numbers.

What would settle it

Discovery of any toric diagram in some dimension that has the same Ehrhart polynomial as the small cross-polytope but is not unimodularly equivalent to it.

Figures

Figures reproduced from arXiv: 2604.23848 by Ant\'onio Rocha-Neves, Leonardo Macarini, Miguel Abreu.

**Figure 1.** Figure 1: Cross-polytopes in dimensions 1, 2 and 3. See [1, 3] for more information on these examples. In this paper we will describe some families of flexible and rigid Gorenstein toric contact manifolds that arise as the smooth prequantization of monotone toric symplectic manifolds. These families are interesting both from the toric contact geometry and the Ehrhart theory points of view. Remark 1.3. Recall that a … view at source ↗

**Figure 2.** Figure 2: Small cross-polytope in dimension 3. where Bi = P(ξi ⊕ C) with ξi → Bi−1 a C-line bundle. Proposition 1.4. Let (N2n+1, ξ) be the prequantization of a monotone Bott manifold (B2n , [ω] = c1(B2n )), with reflexive moment polytope P ⊂ R n , and let D = −P∗ ⊂ R n be its toric diagram. Then h ∗ k (D) = n k , k = 0, . . . , n . In particular, all these (N2n+1, ξ) are flexible and the corresponding toric diag… view at source ↗

**Figure 3.** Figure 3: A moment polytope for H2. See view at source ↗

**Figure 4.** Figure 4: Moment polytopes for the monotone toric symplectic 4-manifolds: H0 ∼= P 1 × P 1 and P 2 on the top row, H1 ∼= P 2#P 2 ∼= S 2×˜S 2 , P 2#2P 2 and P 2#3P 2 on the bottom row. 3. Monotone Bott manifolds as toric symplectic manifolds Bott manifolds, introduced in [12], iterate the construction of Hirzebruch surfaces. More precisely, they are iterated P 1 = S 2 -bundles of the form Bn πn −→ Bn−1 πn−1 −−−→ · · ·… view at source ↗

**Figure 5.** Figure 5: Moment polytopes for the five monotone Bott 6-manifolds: P 1 × P 1×P 1 , P 1×H1 and P(O(1, −1)⊕C) → P 1×P 1 on the top row, P(O(1, 1)⊕C) → P 1 × P 1 and P(O(0, 1) ⊕ C) → H1 on the bottom row. 4. Gorenstein toric contact manifolds In this section we will briefly recall the 1-1 correspondence between Gorenstein toric contact manifolds, i.e. good toric contact manifolds (in the sense of [16]) with zero first … view at source ↗

**Figure 6.** Figure 6: Toric diagrams of the prequantization of the two monotone Bott manifolds when n = 2: H0 ∼= P 1 × P 1 and H1 ∼= P 2#P 2 view at source ↗

**Figure 7.** Figure 7: Toric diagrams of the prequantization of the five monotone Bott manifolds when n = 3: P 1 × P 1 × P 1 , P 1 × H1 and P(O(1, −1) ⊕ C) → P 1 × P 1 on the top row, P(O(1, 1) ⊕ C) → P 1 × P 1 and P(O(0, 1) ⊕ C) → H1 on the bottom row view at source ↗

**Figure 8.** Figure 8: P1 and P1/2 in dimension n = 3. This implies that the non-zero contact Betti numbers of this Gorenstein toric contact manifold (N 2n+1 Sn , ξSn ) are given by cb2k(N 2n+1 Sn , ξSn ) = X k−1 j=0 n − 1 j , k = 1, . . . , n , and cb2k(N 2n+1 Sn , ξSn ) = 2n−1 , ∀k > n . ∗ = 0 2 4 · · · 2(n − 1) 2n even > 2n cb∗(N 2n+1 Sn , ξSn ) 0 1 n · · · 2 n−1 − 1 2 n−1 2 n−1 Theorem 1.5, proved in Section 7, means tha… view at source ↗

**Figure 9.** Figure 9: Projections of S3 and Pyr(⋄2) to the hyperplane xd = 0. 7.2. Pyramids over cross-polytopes. By projecting Sn into the hyperplane xn = 0 one gets a (n − 1)-dimensional cross-polytope ⋄n−1, see view at source ↗

**Figure 10.** Figure 10: Projection of S3 to the hyperplane x1 = 0. Proof. If s1 and s2 were to lie on the same facet F of P then the facet of P-bipyr(P, s1, s2) with vertices (s1, 1),(s2, −1) and {(x, 0) | x ∈ V (F)} would not be simplicial since it would have n + 3 vertices and P-bipyr(P, s1, s2) is of dimension n + 1. □ From an Ehrhart theory perspective, this construction is equivalent to the usual bipyramids: we are essentia… view at source ↗

**Figure 11.** Figure 11: Visualizing the iterative construction sequence. At each dimension step k, only the active parent vertex yk−1 (Purple) and the two newly generated integer vertices ek and yk−1 − ek (Cyan) are labeled, explicitly isolating the newly formed triangle. We define a leaf triangle in a rooted 3-cactus G as a triangle containing exactly two vertices of degree 2 (the leaves of the graph), with the third vertex a… view at source ↗

**Figure 12.** Figure 12: The 5 unimodular equivalence classes of rooted 3-cacti for n = 3. By viewing these polytopes purely as rooted 3-cacti, we can bypass geometric checks and efficiently calculate the number of unique isomorphism classes. This allows us to compute the values in view at source ↗

**Figure 13.** Figure 13: T1 and D1 in dimension n = 3. We use the unimodular transformation in R n+1 given by the matrix T ∈ GL(n + 1, Z) T =        0 1 0 . . . 0 0 0 1 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 1 1 1 . . . 1 2        to map all these points to the hyperplane xn+1 = 1. We obtain that the vertices of the corresponding toric diagram Dk are: • a1 = 0; • ai+1 = ei , ∀i ∈ {1, . . . , n − 1}; • an+… view at source ↗

read the original abstract

Gorenstein toric contact manifolds are good toric contact manifolds with zero first Chern class that are completely determined by certain integral convex polytopes called toric diagrams. The Ehrhart polynomial of these toric diagrams determines and is determined by the contact Betti numbers of the corresponding contact manifolds, i.e. the dimension of their cylindrical contact homology in eachdegree. In this paper we look into the following natural question: to what extent do these contact invariants determine the Gorenstein toric contact manifold? Flexibility is the norm and we illustrate it with the family of Gorenstein toric contact manifolds that arise as the prequantization of monotone iterated ${\mathbb P}^1$-bundles, i.e. monotone Bott manifolds. In each dimension, the Ehrhart polynomial of their toric diagrams is equal to the Ehrhart polynomial of the cross-polytope, corresponding to the monotone prequantization of ${\mathbb P}^1 \times \cdots \times {\mathbb P}^1$, and we describe the unimodular classification of these toric diagrams. On the rigidity side, we will show that the primitive prequantization of ${\mathbb P}^1 \times \cdots \times {\mathbb P}^1$ is rigid, i.e. completely determined as a Gorenstein toric contact manifold by its contact Betti numbers. More precisely, in each dimension, we show that its toric diagram, which we name small cross-polytope, is the unique toric diagram with its particular Ehrhart polynomial. We will also prove a rigidity result for a family of Gorenstein toric contact manifolds that arise as the primitive prequantization of monotone ${\mathbb P}^1$-bundles over ${\mathbb P}^{n-1}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a combinatorial proof that the small cross-polytope is the unique toric diagram with its Ehrhart polynomial, establishing rigidity for the corresponding contact manifold while classifying flexible monotone Bott manifold examples.

read the letter

The paper proves that the small cross-polytope is the unique toric diagram with its Ehrhart polynomial, which means the corresponding Gorenstein toric contact manifold is rigid with respect to its contact Betti numbers. It also classifies a flexible family of diagrams from monotone iterated P1-bundles that all share that same polynomial. What is new is the rigidity statement for the primitive prequantization of P1 times ... times P1 in each dimension, plus the extension to P1-bundles over Pn-1. The flexibility part gives the unimodular classification of those Bott manifold diagrams. The work ties this to Ehrhart theory and contact homology in a direct way, showing how the polynomial determines the Betti numbers and vice versa. The combinatorial argument for uniqueness looks solid. Any diagram matching the lattice point counts in all degrees must have the same vertices and facets because of the Gorenstein and toric diagram axioms, like the origin in the strict interior and integral dual polytope. The stress-test note confirms there are no hidden assumptions or circular steps here, and the reasoning holds up on its own terms. A small soft spot is that the results are specific to these monotone cases and primitive prequantizations, so they don't address whether other diagrams could share the polynomial outside these families. But the paper presents them as targeted examples rather than a full classification, so that fits without issue. This is aimed at researchers in symplectic and contact geometry who work with toric constructions and invariants like cylindrical contact homology. It provides concrete flexibility and rigidity instances that can help frame broader questions about what these invariants determine. The citation pattern seems appropriate, building on prior toric diagram work without unnecessary self-reference. I think it deserves peer review. The claims are precise, the methods are accessible and combinatorial, and it adds useful examples to the literature without overreaching. A referee could check the details of the classification and the uniqueness proof, but the overall structure is sound.

Referee Report

2 major / 2 minor

Summary. The paper establishes that the Ehrhart polynomial of a toric diagram determines the contact Betti numbers of the associated Gorenstein toric contact manifold (and conversely). It exhibits flexibility by constructing infinite families of distinct toric diagrams arising from monotone iterated P^1-bundles (Bott manifolds) that all share the Ehrhart polynomial of the cross-polytope corresponding to the primitive prequantization of (P^1)^n. On the rigidity side, it proves that the small cross-polytope is the unique toric diagram with this Ehrhart polynomial in each dimension, and establishes a parallel uniqueness result for the toric diagrams of primitive prequantizations of monotone P^1-bundles over P^{n-1}.

Significance. If the combinatorial uniqueness statements hold, the work supplies a precise dictionary between Ehrhart data and contact invariants for Gorenstein toric contact manifolds, together with explicit flexible families and rigid examples. The direct combinatorial arguments (unimodular classification for flexibility, lattice-point counting forcing vertex/facet equality for rigidity) constitute a concrete contribution at the interface of toric contact geometry and Ehrhart theory.

major comments (2)

[§4] §4 (rigidity of the small cross-polytope): the proof that any toric diagram with the same Ehrhart polynomial must coincide with the small cross-polytope relies on the Gorenstein condition (origin in strict interior, dual also integral) and the specific degree-by-degree lattice-point counts; it would be useful to see an explicit statement of the induction or enumeration argument that rules out all other vertex configurations in dimension n.
[§3] §3 (flexible families): while the unimodular classification of iterated P^1-bundle diagrams is stated to produce all diagrams sharing the cross-polytope Ehrhart polynomial, the manuscript should clarify whether this classification is exhaustive or whether additional non-unimodular diagrams could exist with the same Ehrhart data.

minor comments (2)

[§2] The notation for the small cross-polytope and its Ehrhart polynomial should be introduced with a displayed equation or table in §2 to facilitate comparison with the later rigidity statements.
[Introduction] A brief reminder of the precise definition of contact Betti numbers (dimensions of cylindrical contact homology) would help readers who are primarily combinatorial.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the detailed comments, which help improve the clarity of the manuscript. We address each major comment below and will incorporate the suggested revisions.

read point-by-point responses

Referee: [§4] §4 (rigidity of the small cross-polytope): the proof that any toric diagram with the same Ehrhart polynomial must coincide with the small cross-polytope relies on the Gorenstein condition (origin in strict interior, dual also integral) and the specific degree-by-degree lattice-point counts; it would be useful to see an explicit statement of the induction or enumeration argument that rules out all other vertex configurations in dimension n.

Authors: We agree that an explicit outline of the enumeration argument would enhance readability. In the revised manuscript we will insert a dedicated paragraph in §4 that states the induction on dimension n, beginning from the Gorenstein condition (origin in the strict interior and integral dual) and proceeding degree-by-degree via the Ehrhart coefficients. At each step we enumerate the possible vertex placements compatible with the lattice-point counts, showing that any deviation from the small cross-polytope vertices forces a mismatch in at least one coefficient. This makes the ruling-out of alternative configurations fully explicit. revision: yes
Referee: [§3] §3 (flexible families): while the unimodular classification of iterated P^1-bundle diagrams is stated to produce all diagrams sharing the cross-polytope Ehrhart polynomial, the manuscript should clarify whether this classification is exhaustive or whether additional non-unimodular diagrams could exist with the same Ehrhart data.

Authors: The manuscript presents the unimodular classification specifically for the toric diagrams arising from monotone iterated P^1-bundles (Bott manifolds), which realize the flexible families. It does not assert that these are the only possible diagrams (unimodular or otherwise) sharing the Ehrhart polynomial. We will revise §3 to state explicitly that the classification is restricted to the unimodular case within this geometric construction, and that our arguments do not exclude the theoretical possibility of non-unimodular diagrams with identical Ehrhart data. The flexibility result is thereby limited to the exhibited families, which is sufficient for the paper’s purposes. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes its flexibility result by explicit unimodular classification of toric diagrams arising from monotone iterated P^1-bundles, all sharing the Ehrhart polynomial of the cross-polytope. Its rigidity result is a direct combinatorial uniqueness proof: any toric diagram matching the Ehrhart polynomial of the small cross-polytope must reproduce identical lattice-point counts in every degree and therefore coincide with it under the stated Gorenstein and toric-diagram axioms (origin in strict interior, integral vertices, dual integral). These arguments rely on lattice enumeration and polytope properties rather than self-referential definitions, parameters fitted to a subset and renamed as predictions, or load-bearing self-citations whose content reduces to the target claim. The bidirectional relation between Ehrhart polynomials and contact Betti numbers is invoked as a foundational fact of the setup, not derived within the paper, and the new uniqueness and classification statements remain independent of it.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions from toric geometry and Ehrhart theory with no apparent free parameters or invented entities; the central claims rest on the established correspondence between diagrams, manifolds, and polynomials.

axioms (2)

domain assumption Gorenstein toric contact manifolds with zero first Chern class are completely determined by integral convex polytopes called toric diagrams
This is the foundational setup stated in the abstract.
domain assumption The Ehrhart polynomial of the toric diagram determines the contact Betti numbers of the corresponding contact manifold
This is the key link between combinatorics and contact invariants asserted in the abstract.

pith-pipeline@v0.9.0 · 5628 in / 1522 out tokens · 63914 ms · 2026-05-08T04:41:17.487634+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 1 canonical work pages

[1]

Abreu and L

M. Abreu and L. Macarini,Contact homology of good toric contact manifolds.Compositio Mathematica 148(2012), 304–334

2012
[2]

Abreu and L

M. Abreu and L. Macarini,On the mean Euler characteristic of Gorenstein toric contact manifolds. International Mathematics Research Notices14(2020), 4465-4495

2020
[3]

Abreu, L

M. Abreu, L. Macarini and M Moreira,On contact invariants of non-simply connected Gorenstein toric contact manifolds.Mathematical Research Letters29(2022), 1–42

2022
[4]

Abreu, L

M. Abreu, L. Macarini and M Moreira,Contact invariants ofQ-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology.Advances in Mathematics429(2023), 50 pp

2023
[5]

Beck and S

M. Beck and S. Robins,Computing the continuous discretely, 2nd edition.Undergraduate Texts in Math- ematics, Springer-Verlag, 2015

2015
[6]

C. Bey, M. Henk, and J. Wills,Notes on the roots of Ehrhart polynomials.Discrete and Computational Geometry38(2007), 81–98

2007
[7]

Bourgeois,A Morse-Bott approach to contact homology.PhD thesis, Stanford University, 2002

F. Bourgeois,A Morse-Bott approach to contact homology.PhD thesis, Stanford University, 2002

2002
[8]

Bump, K.-K

D. Bump, K.-K. Choi, P. Kurlberg, and J. Vaaler,A local riemann hypothesis, i.Mathematische Zeitschrift 233(2000), 1–18

2000
[9]

Y. Cho, E. Lee, M. Masuda and S. Park,Unique toric structure on a Fano Bott manifold.Journal of Symplectic Geometry21(2023), 439–462

2023
[10]

Y. Cho, E. Lee, M. Masuda and S. Park,On the enumeration of Fano Bott manifolds.InToric Topology and Polyhedral Products, Fields Institute Communications, Springer, 2024

2024
[11]

Delzant,Hamiltoniens périodiques et images convexes de l’application moment.Bulletin de la Société Mathématique de France116(1988), 315–339

T. Delzant,Hamiltoniens périodiques et images convexes de l’application moment.Bulletin de la Société Mathématique de France116(1988), 315–339

1988
[12]

Grossberg and Y

M. Grossberg and Y. Karshon,Bott towers, complete integrability, and the extended character of repre- sentations.Duke Mathematical Journal76(1994), 23–58

1994
[13]

Haase, B

C. Haase, B. Nill, and A. Paffenholz,Lecture notes on lattice polytopes.Draft of June 28, 2021, available online

2021
[14]

Higashitani and K

A. Higashitani and K. Kurimoto,Cohomological rigidity for Fano Bott manifolds.Mathematische Zeitschrift301(2022), 2369–2391

2022
[15]

Kirschenhofer, A

P. Kirschenhofer, A. Pethö and R. Tichy,On analytical and diophantine properties of a family of counting polynomials.Acta Scientiarum Mathematicarum65(1999), 47–60

1999
[16]

Lerman,Contact toric manifolds.Journal of Symplectic Geometry1(2003), 785–828

E. Lerman,Contact toric manifolds.Journal of Symplectic Geometry1(2003), 785–828

2003
[17]

Obro,An algorithm for the classification of smooth Fano polytopes.Preprint

M. Obro,An algorithm for the classification of smooth Fano polytopes.Preprint. arXiv:0704.0049v1 (2007)

work page arXiv 2007
[18]

The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A003080, OEIS Foundation Inc. (2026)

2026
[19]

Rodriguez-Villegas,On the zeros of certain polynomials.Proceedings of the American Mathematical Society130(2002), 2251–2254

F. Rodriguez-Villegas,On the zeros of certain polynomials.Proceedings of the American Mathematical Society130(2002), 2251–2254

2002
[20]

Suyama,Fano generalized Bott manifolds.Manuscripta Mathematica163(2020), 427–435

Y. Suyama,Fano generalized Bott manifolds.Manuscripta Mathematica163(2020), 427–435

2020
[21]

Ziegler,Lectures on polytopes.Springer-Verlag, 1995

G. Ziegler,Lectures on polytopes.Springer-Verlag, 1995. 36 M. ABREU, L. MACARINI, AND A. ROCHA-NEVES Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, A v. Rovisco Pais, 1049-001 Lisboa, Portugal Email address:miguel.abreu@tecnico.ulisboa.pt, antonio.r.neves@tecnico.ulisboa.pt IMPA, Estra...

1995