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Degree shifts between q-deformed friezes and q-Farey labelings for general triangulations

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Pith's one-line read For general triangulations the numerator and denominator polynomials of q-deformed continued fractions coincide exactly with entries of q-deformed Conway-Coxeter friezes, while q-Farey polynomials agree up to powers of q counted by the numb

desk verdict The paper extends the q-deformed frieze and Farey correspondences to general triangulations by giving explicit q-power corrections counted by diagonals or 1-entries. read the letter →

arxiv 2605.31333 v1 pith:RK5VU5TK submitted 2026-05-29 math.CO math.QA

Manabu Inuma , Ren Motomiya , Takeyoshi Kogiso This is my paper · ORCID
classification math.COmath.QA
keywords q-deformedcontinuedfractionsq-FareylabelingsfriezesConway-Coxetertriangulationsquidditiespolynomialdegreesdegreeshifts
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The pith

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

The reading

The paper extends the correspondences among q-deformed continued fractions, Farey labelings, and Conway-Coxeter friezes from triangulations with exactly two exterior cells to arbitrary subsequences of quiddities from any triangulation. It proves that the numerator and denominator polynomials of the q-continued fraction equal the corresponding entries in the q-frieze without adjustment. The polynomials arising from the q-Farey labeling match those same values only after multiplication by explicit powers of q, where the exponents are given combinatorially by the number of diagonals in the triangulation or equivalently by the number of 1-entries in the frieze. The work further determines the minimum and maximum degrees of all these polynomials using the same combinatorial count.

What carries the argument

The combinatorial count of diagonals in the triangulation (equivalently the number of 1-entries in the frieze), which supplies the explicit exponents for the degree shifts between the three q-labelings.

What would settle it

Compute the three families of polynomials for one concrete triangulation that has three or more exterior cells, then check whether the degree difference between the Farey and frieze versions equals the number of 1-entries in the frieze.

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Extended reading notes

Core claim

We show that the numerator and denominator polynomials of q-deformed continued fractions coincide with entries of q-deformed Conway-Coxeter friezes, while the corresponding polynomials in q-Farey labelings agree with them up to explicit powers of q. These powers are described combinatorially in terms of the number of diagonals in the triangulation, or equivalently, the number of entries equal to 1 in the associated frieze. Furthermore, we determine the minimum and maximum degrees of these polynomials in terms of the same combinatorial data.

Load-bearing premise

The q-deformations of continued fractions, friezes, and Farey labelings are defined consistently for quiddities coming from any triangulation.

Editorial extensions

If this is right

  • The q-continued-fraction numerator and denominator equal the q-frieze entries with no extra factors.
  • Each q-Farey polynomial equals the matching frieze entry multiplied by q raised to the diagonal count.
  • The lowest and highest degrees of every polynomial are fixed once the number of 1-entries is known.
  • All stated equalities and degree formulas hold for every triangulation without further restrictions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit power-of-q rule supplies a direct translation map between the three labelings that does not require recomputing the underlying continued fraction.
  • The min/max degree formulas give immediate bounds on polynomial size for any triangulation once its 1-entry count is read off.
  • The same combinatorial count may serve as a complexity measure when comparing q-deformed objects attached to different classes of polygons.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper extends q-deformations of continued fractions, Conway-Coxeter friezes, and Farey labelings (originally due to Morier-Genoud and Ovsienko for triangulations with exactly two exterior cells) to arbitrary subsequences of quiddities from general triangulations. It claims that the numerator and denominator polynomials of q-deformed continued fractions coincide with entries of q-deformed friezes, that the corresponding polynomials from q-Farey labelings agree with them up to explicit powers of q (described combinatorially via the number of diagonals or the number of 1-entries in the frieze), and that the minimum and maximum degrees of these polynomials are determined by the same combinatorial data.

Significance. If the extension and the stated equalities hold, the work provides a useful generalization of the q-deformed correspondences, with explicit combinatorial control over degree shifts. This could strengthen connections between q-analogs, friezes, and triangulations in the broader context of cluster algebras and combinatorial representation theory.

major comments (1)
  1. [Introduction and main theorems] The central claims rest on the q-deformations and their recurrence/exchange relations extending without modification or extra correction terms to arbitrary quiddity subsequences arising from triangulations with more than two exterior cells. The manuscript must supply an explicit verification or inductive argument for this extension step, as it is load-bearing for the polynomial coincidences and the combinatorial description of the q-powers.
minor comments (1)
  1. Clarify the precise statement of the prior definitions (from Morier-Genoud-Ovsienko) that are being invoked for the general case, including any restrictions that may or may not carry over.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestion regarding the extension to general triangulations. We address the single major comment below.

read point-by-point responses

  1. Referee: [Introduction and main theorems] The central claims rest on the q-deformations and their recurrence/exchange relations extending without modification or extra correction terms to arbitrary quiddity subsequences arising from triangulations with more than two exterior cells. The manuscript must supply an explicit verification or inductive argument for this extension step, as it is load-bearing for the polynomial coincidences and the combinatorial description of the q-powers.

    Authors: We agree that an explicit verification strengthens the manuscript. The q-deformed recurrences and exchange relations are local (depending only on consecutive quiddity entries and the triangle structure), so they extend verbatim to arbitrary subsequences without correction terms; this is used throughout Sections 2–4 to establish the polynomial identities. To make the step fully self-contained, we will add a short inductive lemma (with base case for two exterior cells and inductive step via diagonal flips) immediately after the definitions in the revised version. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; extension relies on external prior definitions

full rationale

The paper cites Morier-Genoud and Ovsienko (distinct external authors) for the original q-deformations and restricted correspondences, then extends them to general triangulations by showing algebraic coincidences and combinatorial degree shifts. No self-citations appear, no parameters are fitted then renamed as predictions, and no definitions reduce to their own outputs by construction. The work is self-contained against the external benchmarks it invokes, with new combinatorial claims (powers counted by diagonals/1-entries, min/max degrees) that do not tautologically follow from the inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on extending the q-deformations defined in prior literature to general triangulations using the combinatorial structure of quiddities and diagonals; no new free parameters or invented entities are indicated.

assumptions (1)
  • domain assumption q-deformations of continued fractions, Farey labelings, and Conway-Coxeter friezes as introduced by Morier-Genoud and Ovsienko
    The paper extends these established objects to arbitrary triangulations.

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Pith. "Pith review of Degree shifts between q-deformed friezes and q-Farey labelings for general triangulations." pith.science (2026). https://pith.science/paper/RK5VU5TK

@misc{pith2026260531333,
  author       = {Pith},
  title        = {Pith review of: Degree shifts between q-deformed friezes and q-Farey labelings for general triangulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RK5VU5TK}},
  note         = {Machine review of arXiv:2605.31333}
}
read the original abstract

Morier-Genoud and Ovsienko introduced q-deformations of continued fractions, Farey labelings, and Conway--Coxeter friezes, and established relationships among them in restricted settings associated with triangulations having exactly two exterior cells. In this paper, we extend these correspondences to arbitrary subsequences of quiddities arising from general triangulations. We show that the numerator and denominator polynomials of q-deformed continued fractions coincide with entries of q-deformed Conway--Coxeter friezes, while the corresponding polynomials in q-Farey labelings agree with them up to explicit powers of q. These powers are described combinatorially in terms of the number of diagonals in the triangulation, or equivalently, the number of entries equal to 1 in the associated frieze. Furthermore, we determine the minimum and maximum degrees of these polynomials in terms of the same combinatorial data.

Figures

Figures reproduced from arXiv: 2605.31333 by the authors.

Figure 2.1
Figure 2.1. Cell with vertices vα, vβ, vγ Definition 2.2 (Quiddity). For a triangulation T of an n-gon and a vertex vα, a cell C is said to be adjacent to vα if vα is a vertex of C. Let cell(vα) be the set of such cells, and write cα = # cell(vα). We call a sequence (cα)α∈Z a quiddity of T, and define quid(T) = (cα)α∈Z. Note that quid(T) is a cyclic sequence of period n; namely, ci = ci+n for any i ∈ Z. Definition 2.3 (Exterior… view at source ↗
Figure 2.2
Figure 2.2. Descendant degree of vj for a triangulation (T, i) (j ̸= i, i + n − 1) 2.2 Farey labelings Definition 2.10 (Farey sum). Let P 1 (Q) be the union of the set of rational numbers Q and the point at infinity 1 0 : Q ∪  1 0  . We call elements of P 1 (Q) rational numbers. For two rational numbers ρα = rα sα , ρβ = rβ sβ ∈ P 1 (Q), the Farey sum ρα ⊕ ρβ ∈ P 1 (Q) is defined by ρα ⊕ ρβ = rα + rβ sα + sβ . Definition 2.11… view at source ↗
Figure 2.3
Figure 2.3. Conway-Coxeter frieze {σk,ℓ}(i,j)∈D For each (k, ℓ) ∈ D and u ∈ Z, define subsets of D, Dtri k,ℓ, Dmd k,ℓ , Dad k,ℓ, and Drec k,u,ℓ as follows. For (k, ℓ) ∈ D with ℓ − k = 0, 1, set Dtri k,ℓ = Dmd k,ℓ = Dad k,ℓ = Drec k,u,ℓ = ϕ. For (k, ℓ) ∈ D with ℓ − k ≥ 2, define Dtri k,ℓ = {(a, b) ∈ D | k ≤ a, a + 2 ≤ b ≤ ℓ} , Dmd k,ℓ = {(k, k + 2),(k, k + 3), . . . ,(k, ℓ)} = {(k, b) ∈ D | k + 2 ≤ b ≤ ℓ} , Dad k,ℓ = {(k, ℓ),(k … view at source ↗
Figures from the paper (7 more)
Figure 2.4
Figure 2.4. Figure 2.4: Triangulation T with quid(T) = (. . . , 1, 4, 2, 1, 3, 4, 1, 2, 3, . . .) 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: CCF associated with the triangulation T with quid(T) = (. . . , 1, 4, 2, 1, 3, 4, 1, 2, 3, . . .) 2.4 Unimodular matrices Definition 2.17 (Unimodular matrix). Let SL(2, Z) denote the group of 2 × 2 unimodular matrices over Z, that is, SL(2,Z) =  A =  a b c d a, b,…
Figure 3.1
Figure 3.1. Figure 3.1: q-CCF associated with the triangulation T with quid(T) = (. . . , 1, 4, 2, 1, 3, 4, 1, 2, 3, . . .) For example, q-continued fraction [[c3, c4, c5, c6, c7]]q = [[1, 3, 4, 1, 2]]q is computed as follows: [[1, 3, 4, 1, 2]]q = 1 − 1 [3] − q 2 [4] − q 3 1 − 1 [2] = 1 − 1…
Figure 3.2
Figure 3.2. Figure 3.2: Side-weight triple of a cell C Example 3.12. Consider the triangulation in Example 2.16, we take the edge v2v3 = v11v3 as the base edge. The q-Farey labeling Fareyq (T, 3) = (θ3,j )3≤j≤11 of (T, 3) is given in [PITH_FULL_IMAGE:figures/full_fig_p025_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: q-Farey labeling of the triangulation (T, 3) with quiddity quid(T) = (. . . , 1, 4, 2, 1, 3, 4, 1, 2, 3, . . .) The following Proposition 3.13 establishes positivity of the coefficients, as well as unit trailing and leading coefficients for the numerators and denomin…
Figure 3.4
Figure 3.4. Figure 3.4: q-Farey labeling Fareyq (T, 3) and descendant degrees for the triangulation (T, 3) with quiddity quid(T) = (1, 4, 2, 1, 3, 4, 1, 2, 3) From Proposition 3.6, the q-continued fractions [[c3, c4, . . . , cj ]]q for 2 ≤ j ≤ 11 are given by the entries in the associated q…
Figure 5.1
Figure 5.1. Figure 5.1: q-Farey labeling Fareyq (T, 7) and descendant degrees for the triangulation (T, 7) with quiddity quid(T) = (1, 4, 2, 1, 3, 4, 1, 2, 3) As in Example 3.16, Proposition 3.6, and Theorem 3.15 yield the following correspondence between q-continued fractions, the entries …

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Works this paper leans on

31 extracted references · 1 canonical work pages

  1. [1]

    Friezes.Advances in Mathematics, 225(6):3134–3165, 2010

    Ibrahim Assem, Christophe Reutenauer, and David Smith. Friezes.Advances in Mathematics, 225(6):3134–3165, 2010

    2010

  2. [2]

    Asilata Bapat, Louis Becker, and Anthony M Licata.q-deformed rational numbers and the 2-Calabi– Yau category of typeA 2.Forum of Mathematics, Sigma, 11:e47, 41 pp., 2023

    2023

  3. [3]

    Cluster algebras and continued fractions.Compositio mathematica, 154(3):565–593, 2018

    ˙Ilke C ¸ anak¸ cı and Ralf Schiffler. Cluster algebras and continued fractions.Compositio mathematica, 154(3):565–593, 2018

    2018

  4. [4]

    Quiddities of polygon dissections and the Conway–Coxeter frieze equation.ANNALI SCUOLA NORMALE SUPERIORE-CLASSE DI SCIENZE, 24(4):2125– 2170, 2023

    Charles H Conley and Valentin Ovsienko. Quiddities of polygon dissections and the Conway–Coxeter frieze equation.ANNALI SCUOLA NORMALE SUPERIORE-CLASSE DI SCIENZE, 24(4):2125– 2170, 2023

    2023

  5. [5]

    Triangulated polygons and frieze patterns.The Mathemat- ical Gazette, 57(400):87–94, 1973

    John H Conway and Harold SM Coxeter. Triangulated polygons and frieze patterns.The Mathemat- ical Gazette, 57(400):87–94, 1973

    1973

  6. [6]

    Triangulated polygons and frieze patterns, continued.The Mathematical Gazette, 57(401):175–183, 1973

    John H Conway and Harold SM Coxeter. Triangulated polygons and frieze patterns, continued.The Mathematical Gazette, 57(401):175–183, 1973

    1973

  7. [7]

    Frieze patterns.Acta Arithmetica, 18(1):297–310, 1971

    Harold SM Coxeter. Frieze patterns.Acta Arithmetica, 18(1):297–310, 1971

    1971

  8. [8]

    A quantum Teichm¨ uller space.Theoretical and Mathematical Physics, 120(3):1245–1259, 1999

    Vladimir V Fock and Leonid O Chekhov. A quantum Teichm¨ uller space.Theoretical and Mathematical Physics, 120(3):1245–1259, 1999

    1999

Show all 31 references
  1. [9]

    Cluster algebras I: foundations.Journal of the American mathematical society, 15(2):497–529, 2002

    Sergey Fomin and Andrei Zelevinsky. Cluster algebras I: foundations.Journal of the American mathematical society, 15(2):497–529, 2002

    2002

  2. [10]

    Cluster algebras II: Finite type classification.Inventiones mathematicae, 154:63–121, 2002

    Sergey Fomin and Andrei Zelevinsky. Cluster algebras II: Finite type classification.Inventiones mathematicae, 154:63–121, 2002

    2002

  3. [11]

    Cluster algebras IV: coefficients.Compositio Mathematica, 143(1):112–164, 2007

    Sergey Fomin and Andrei Zelevinsky. Cluster algebras IV: coefficients.Compositio Mathematica, 143(1):112–164, 2007

    2007

  4. [12]

    Takeyoshi Kogiso.q-deformations andt-deformations of Markov triples, 2020.arXiv:2008.12913

    2020

  5. [13]

    Arithmetic onq-deformed rational numbers.Arnold Mathematical Journal, 11(3):42–92, 2025

    Takeyoshi Kogiso, Kengo Miyamoto, Xin Ren, Michihisa Wakui, and Kohji Yanagawa. Arithmetic onq-deformed rational numbers.Arnold Mathematical Journal, 11(3):42–92, 2025

    2025

  6. [14]

    A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials.Journal of Knot Theory and Its Ramifications, 28(14):article no

    Takeyoshi Kogiso and Michihisa Wakui. A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials.Journal of Knot Theory and Its Ramifications, 28(14):article no. 1950083, 40 pp., 2019

    2019

  7. [15]

    Theq-analog of the Markoff injectivity conjecture over the language of a balanced.Combinatorial Theory, 2(1):article no

    S´ ebastien Labb´ e and M´ elodie Lapointe. Theq-analog of the Markoff injectivity conjecture over the language of a balanced.Combinatorial Theory, 2(1):article no. 9, 25 pp., 2022. 42

    2022

  8. [16]

    102223, 28 pp., 2021

    Ludivine Leclere and Sophie Morier-Genoud.q-deformations in the modular group and of the real quadratic irrational numbers.Advances in Applied Mathematics, 130:article no. 102223, 28 pp., 2021

    2021

  9. [17]

    On radius of convergence ofq-deformed real numbers.Moscow Mathematical Journal, 24(1):1–19, 2024

    Ludivine Leclere, Sophie Morier-Genoud, Valentin Yur’evich Ovsienko, and Aleksandr Petrovich Veselov. On radius of convergence ofq-deformed real numbers.Moscow Mathematical Journal, 24(1):1–19, 2024

    2024

  10. [18]

    Cluster algebras and Jones polynomials.Selecta Mathematica, 25(4):article article no

    Kyungyong Lee and Ralf Schiffler. Cluster algebras and Jones polynomials.Selecta Mathematica, 25(4):article article no. 58, 41 pp., 2019

    2019

  11. [19]

    On a rank-unimodality conjecture of Morier-Genoud and Ovsienko.Discrete Mathematics, 344(8):article no

    Thomas McConville, Bruce E Sagan, and Clifford Smyth. On a rank-unimodality conjecture of Morier-Genoud and Ovsienko.Discrete Mathematics, 344(8):article no. 112483, 13 pp., 2021

    2021

  12. [20]

    Farey boat: continued fractions and triangulations, modular group and polygon dissections.Jahresbericht der Deutschen Mathematiker-Vereinigung, 121:91–136, 2019

    Sophie Morier-Genoud and Valentin Ovsienko. Farey boat: continued fractions and triangulations, modular group and polygon dissections.Jahresbericht der Deutschen Mathematiker-Vereinigung, 121:91–136, 2019

    2019

  13. [21]

    In Forum of Mathematics, Sigma, volume 8, page e13, 2020

    Sophie Morier-Genoud and Valentin Ovsienko.q-deformed rationals andq-continued fractions. In Forum of Mathematics, Sigma, volume 8, page e13, 2020

    2020

  14. [22]

    Quantum numbers andq-deformed Conway–Coxeter friezes.The Mathematical Intelligencer, 43(2):61–70, 2021

    Sophie Morier-Genoud and Valentin Ovsienko. Quantum numbers andq-deformed Conway–Coxeter friezes.The Mathematical Intelligencer, 43(2):61–70, 2021

    2021

  15. [23]

    Sl 2(Z)-tilings of the torus, Coxeter–Conway friezes and Farey triangulations.L’Enseignement Math´ ematique, 61(1):71–92, 2015

    Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov. Sl 2(Z)-tilings of the torus, Coxeter–Conway friezes and Farey triangulations.L’Enseignement Math´ ematique, 61(1):71–92, 2015

    2015

  16. [24]

    Cluster variables, ancestral triangles and Alexander polynomials

    Wataru Nagai and Yuji Terashima. Cluster variables, ancestral triangles and Alexander polynomials. Advances in Mathematics, 363:article no. 106965, 37 pp., 2020

    2020

  17. [25]

    Oriented posets, rank matrices andq-deformed Markov numbers.Discrete Mathematics, 348(2):article no

    Ezgi Kantarcı O˘ guz. Oriented posets, rank matrices andq-deformed Markov numbers.Discrete Mathematics, 348(2):article no. 114256, 17 pp., 2025

    2025

  18. [26]

    Rank polynomials of fence posets are unimodal

    Ezgi Kantarcı O˘ guz and Mohan Ravichandran. Rank polynomials of fence posets are unimodal. Discrete Mathematics, 346(2):article no. 113218, 20 pp., 2023

    2023

  19. [27]

    Towards quantized complex numbers:q-deformed gaussian integers and the picard group.Open Communications in Nonlinear Mathematical Physics, 1:73–93, 2021

    Valentin Ovsienko. Towards quantized complex numbers:q-deformed gaussian integers and the picard group.Open Communications in Nonlinear Mathematical Physics, 1:73–93, 2021

    2021

  20. [28]

    F-polynomial formula from continued fractions.Journal of Algebra, 509:467–475, 2018

    Michelle Rabideau. F-polynomial formula from continued fractions.Journal of Algebra, 509:467–475, 2018

    2018

  21. [29]

    On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Research in Number Theory, 8(3):article no

    Xin Ren. On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Research in Number Theory, 8(3):article no. 37, 14 pp., 2022

    2022

  22. [30]

    Corrigendum to: On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Research in Number Theory, 9(2):article no

    Xin Ren. Corrigendum to: On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Research in Number Theory, 9(2):article no. 39, 4 pp., 2023

    2023

  23. [31]

    The modular surface and continued fractions.Journal of the London Mathematical Society, 2(1):69–80, 1985

    Caroline Series. The modular surface and continued fractions.Journal of the London Mathematical Society, 2(1):69–80, 1985. 43

    1985

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