Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Cluster algebraic interpretation of generalized Markov numbers and their matrixizations

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Generalized Markov numbers are shown to be specializations of order-ideal sums over fence posets, and the same posets yield explicit 2x2 cluster matrices for every rational label.

desk verdict A substantial new unification of the cluster and matrix approaches to generalized Markov numbers, with a main theorem that is likely true but rests on two under-verified inputs that should be tightened before acceptance. read the letter →

arxiv 2507.06900 v2 pith:KPW5RUE3 submitted 2025-07-09 math.CO math.NT

Esther Banaian , Yasuaki Gyoda This is my paper · ORCID
classification math.COmath.NT MSC 13F6011D2511A55
keywords MarkovnumbersgeneralizedequationclusteralgebrasCohnmatricesMarkov-monodromyfenceposetsorderidealsskeinrelations
verification ladder T0 review T1 audit T2 compute T3 formal

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 tries to show that generalized Markov numbers—integer solutions of $x^2+y^2+z^2+k_1yz+k_2xz+k_3xy=(3+k_1+k_2+k_3)xyz$—are governed by one combinatorial object: a fence poset attached to each rational $p/q$. It claims that in the generalized Markov cluster algebra $A(k_1,k_2,k_3)$, every cluster variable in the $k_1$-branch is an explicit positive Laurent polynomial computed by summing weights of order ideals in that poset, and that the same poset data assemble into two families of $2\times2$ matrices whose $(1,2)$-entries are those cluster variables. Substituting $x_1=x_2=x_3=1$ turns the matrices into previously studied integer matrices carrying generalized Markov numbers, so the paper unifies the cluster-algebraic and matrix approaches to Markov numbers. It also classifies all triples of such matrices by binary trees and proves a new skein-type identity for the poset polynomials.

What carries the argument

The central object is the labeled, weighted fence poset $P_{p/q}$, produced by a construction algorithm that follows the segment $\gamma_{p/q}$ through the lattice and labels each element by one of $x_1,x_2,x_3$ with weights built from the variables $\hat{x}_i$ and the parameters $k_i$. The carrying identity is the order-ideal expansion $x_{1,p/q}=x^{g_{p/q}}W(P_{p/q})$, where $W(P)$ is the sum of products of weights over all order ideals; the proof runs by induction along Farey triples using the generalized F-polynomial recurrence, a crossing-overlap skein relation (Proposition 8.4), a new reverse-kissing self-overlap relation (Proposition 8.12), and explicit c-vector formulas. The matrices $C_{p/q}$ and $M_{p/q}$ are assembled directly from these $W$-polynomials and g-vector monomials, and their triples propagate by the operations $Q\mapsto PQ-S_R$ for cluster generalized Cohn matrices and by conjugation for cluster Markov-monodromy matrices.

What would settle it

Compute the principal-coefficient cluster variable $x^{\mathrm{prin}}_{1,2/3}$ in $A(1,1,1)$ by direct generalized mutation from the initial seed and compare it, order ideal by order ideal, with $x^{g_{2/3}} W^{\mathrm{prin}}(P_{2/3})$; any mismatch would disprove the central expansion theorem. A second decisive test is to search the mutation tree of $A(1,2,3)$ for a cluster variable that appears in two different positions, which would refute Theorem 3.3 and with it the well-definedness of every matrix in the paper.

Watch

Extended reading notes

Core claim

The central claim is Theorem 8.30: with principal coefficients, the cluster variable labeled by $p/q$ in the $k_1$-branch equals $x^{g_{p/q}} W^{\mathrm{prin}}(P_{p/q})$, where $P_{p/q}$ is a fence poset built from the straight segment $\gamma_{p/q}$ crossing a lattice of slope $0,\infty$ and $-1$ lines, and $W^{\mathrm{prin}}$ sums the weights of its order ideals. The coefficient-free version (Corollary 8.31) says the same with $y_i$ set to $1$, so each generalized Markov number is the specialization of such a sum. Theorem 9.3 packages this into matrices: for each Farey triple $(p/q,(p+r)/(q+s),r/s)$, the matrices $C_{p/q}$ are $(k_1,k_2,k_3)$-cluster generalized Cohn matrices, lie in $SL(2,\mathbb{Z}[x_1^{\pm1},x_2^{\pm1},x_3^{\pm1}])$, have $(1,2)$-entry $x_{1,p/q}$, and satisfy the trace identity $\mathrm{tr}(C_{p/q})=M x_{1,p/q}-k_{p/q}$. Theorem 10.1 gives the companion cluster Markov-monodromy matrices $M_{p/q}$ obtained by a fixed conjugation, and the two families are connected by explicit tree isomorphisms. The result is that every cluster variable in the branch and every generalized Markov triple in the corresponding tree carries an explicit order-ideal expansion and an explicit matrix realization.

Load-bearing premise

The load-bearing premise is that no cluster variable shows up in two different slots of the clusters, so every variable has a unique rational label and a definite parity $k_{p/q}$; the paper's proof of this uniqueness is informal and depends on the assertion that once a variable leaves the cluster tree it never returns in another position. If a variable reappeared in a different slot, the parity would be undefined and every matrix trace condition and the whole poset construction would collapse.

Editorial extensions

If this is right

  • Every cluster variable in the $k_1$-branch of $A(k_1,k_2,k_3)$ has an explicit positive Laurent expansion whose terms are in bijection with order ideals of a fence poset; setting $x_i=1$ recovers generalized Markov numbers.
  • For every Farey triple, the matrices $C_{p/q}$ and $M_{p/q}$ form cluster generalized Cohn and cluster Markov-monodromy triples, giving explicit $SL(2,\mathbb{Z}[x^{\pm}])$ realizations whose $(1,2)$-entries are the cluster variables in that triple.
  • Specializing $x_1=x_2=x_3=1$ reproduces the integer generalized Cohn and Markov-monodromy matrices of prior work, showing that the integer matrices are specializations of the cluster matrices.
  • The classification theorems for CGC and CMM triples show that all such triples, not just the combinatorial ones, occur in the binary trees $CGCT$ and $CMMT$, with the combinatorial tree forming one explicit branch.
  • The new skein relation yields identities among poset weight polynomials that mimic relations among Caldero-Chapoton functions of string modules, giving a byproduct about such functions.

Reading between the lines

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

  • By the symmetry of the construction, the same poset expansion should hold for the $k_2$- and $k_3$-branches after permuting indices; the paper states the $k_1$-branch and notes the extension but does not spell it out.
  • The explicit order-ideal formula suggests a direct counting algorithm for generalized Markov numbers and may provide a concrete way to test uniqueness-type questions for fixed $(k_1,k_2,k_3)$ by inspecting the posets.
  • The reverse self-overlap identity may generalize to a family of algebraic relations indexed by closed curves or band modules, connecting the poset calculus to orbifold skein algebras.
  • Because the matrices are explicit in the initial variables, the tree of $C_{p/q}$ could serve as a matrix-valued continued fraction for rational labels, potentially yielding new Diophantine approximations tied to $(k_1,k_2,k_3)$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces two families of SL(2, Z[x1^±, x2^±, x3^±]) matrices associated to the generalized Markov cluster algebras A(k1,k2,k3): cluster generalized Cohn (CGC) matrices and cluster Markov-monodromy (CMM) matrices. It classifies all CGC and CMM triples by binary trees (Theorems 4.13, 4.16, 5.9, 5.12), constructs an isomorphism between the CGC and CMM tree structures, and then gives an explicit combinatorial family for one tree. The explicit family is based on weighted fence posets P_{p/q}: Theorem 8.30 asserts that the order-ideal generating function of P_{p/q} is exactly the principal-coefficient cluster variable x^{prin}_{1,p/q}, and Theorem 9.3 assembles these poset formulas into CGC matrices and CGC triples indexed by Farey triples. Theorem 10.1 transfers the construction to CMM matrices via the map ψ^{-1}_M. The final section compares the new matrices with integer Cohn matrices, q-deformations, and snake-graph matrix formulas.

Significance. If the main results are correct, the paper provides a genuinely explicit order-ideal expansion for every cluster variable in the k1-branch of a broad family of generalized cluster algebras, and it connects this expansion to Cohn-type and Markov-monodromy matrices in SL(2, Z[x_i^±]). The matrix classification theorems give a complete tree-theoretic description of the two matrix families, and the poset machinery produces a new skein-like relation for Caldero-Chapoton-type generating functions. The paper is strong on concreteness: the poset construction is algorithmic, the base cases of the induction in Theorem 8.30 are checked directly, and the matrix identities in Sections 9 and 10 are verified entry by entry using the skein relations. The main risk is not the overall architecture but two load-bearing technical inputs—the position-uniqueness/parity theorem and the c-vector rescaling lemma—whose proofs in the manuscript are compressed; these feed directly into equations (8.2)-(8.3) of the central induction.

major comments (3)
  1. [§3.4, proof of Theorem 3.3] The proof of Theorem 3.3 is one paragraph and informal: it asserts, using Proposition 3.8 and Corollary 3.23, that variables which disappear under mutation away from the root never reappear, so each variable has a unique position and parity. This is load-bearing because the definition of k_x, the degree d_{p/q} in equations (8.2)-(8.3), and Lemma 8.20 all depend on well-definedness of the parity. Moreover, Corollary 3.23 depends on Proposition 2.11, which is imported from [43] and only asserted to generalize to generalized cluster algebras via [7, Theorems 7.9 and 8.13]. Please supply a complete proof of Theorem 3.3, or at minimum a detailed induction showing injectivity of the fraction labeling into the set of cluster variables before Corollary 3.23 is used.
  2. [§3.4, Lemma 3.25] Lemma 3.25 is stated with a proof sketch: after invoking [42, Proposition 3.21] and the identity for skew-symmetric Markov matrices, the c-vector rescaling c_{ij;t} = (d_i/d_j) c^M_{ij;t} is deduced from a conjugation by a diagonal matrix. The exact convention for C-matrix versus G-matrix duality is delicate, and the choices of R in the two nonscalar cases are not fully written out. Since the y-monomial exponents in the F-polynomial induction of Theorem 8.30 are precisely these rescaled c-vectors, a sign or ordering error here would break the comparison with the poset skein relations. Please write out the verification for each of the four cases (I)-(IV), including the full diagonal conjugation, or provide an independent computational check for several small Farey triples.
  3. [§3.4, proof of Theorem 3.21] The proof of Theorem 3.21 says that 'for all possible matrices B, DB is equal to the matrix associated to the Markov cluster algebra.' In cases (II) and (III), D is not scalar and this is false; what is true is that BD equals the Markov exchange matrix, which is the matrix appearing in Proposition 2.8 for g-vectors. This appears to be a typo, but since Theorem 3.21 feeds into Corollary 3.23 and hence Theorem 3.3, the statement and proof should be corrected explicitly so that the intended use of Proposition 2.8 is unambiguous.
minor comments (4)
  1. [§4, Theorem 4.12] Theorem 4.12 is stated without proof, with only the comment that the proof method is the same as Theorem 4.6. Given that the inverse tree classification in Theorem 4.13 and the CMM comparison rely on this statement, it would be helpful to include the analogues of Lemmas 4.8-4.10 or at least a precise sentence explaining which identities are reused.
  2. [Throughout] There are several typographical errors: 'Markov-Monodoromy' in the title of Sections 5 and 10, 'sytstem' in Proposition 5.3, 'breif' in the introduction, and 'combinatorical' in Example 10.2. These should be corrected before publication.
  3. [§9, Proposition 9.10] The proof of Proposition 9.10 invokes [26, Proposition 4.2] for the matrix product structure. Since that reference is used for an essential entry-wise comparison, it would be helpful to state explicitly which of its formulas are being used and how the leading monomials x^g are matched.
  4. [§11.1] The comparison with the integer-entry matrices of [23] is stated in terms of replacing pairs of weight k, 1/k by chains of k+1 elements. This is plausible but is asserted rather than proved; a short example or explicit reference to the relevant theorem in [23] would make the comparison easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the poset expansion is proved by matching the cluster-algebra recursion using independent c-vector and g-vector facts, with no fitted parameter renamed as a prediction.

full rationale

The central claim, Theorem 8.30, is not equivalent to its own inputs. The poset P_{p/q} is constructed from the line segment gamma_{p/q} and from label/weight data determined by the k_i, independently of the cluster variable x_{1,p/q}. The g-vector of the poset is computed combinatorially in Lemma 7.5 and then equated with the known cluster g-vector from Theorem 3.21, which is imported from Nájera Chávez's Markov cluster algebra result and Nakanishi's generalized cluster algebra results; these are external, parameter-free facts. The proof of Theorem 8.30 then verifies that W^{prin}(P_{p/q}) satisfies exactly the same F-polynomial mutation identities (8.2) and (8.3), with the c-vectors supplied by Proposition 3.24 and Lemma 3.25, and with base cases checked directly. No parameter is fitted to the target identity, and no cluster variable is defined in terms of the poset. The later matrix theorems (9.3 and 10.1) derive determinant and trace identities from the same expansion, and the specialization at x_i = 1 recovers previously known Cohn and Markov-monodromy matrices, providing an external benchmark. The position-uniqueness result, Theorem 3.3, is proved informally and would benefit from additional verification, but it is justified via g-vector separation from Proposition 2.11 rather than by assuming the conclusion; this is a rigor concern, not a circular one. The self-citations to [2] and [3] are supporting prior results, such as a weighted-poset partition in [2], with stated assumptions that do not include the target theorem; they are not invoked as uniqueness theorems to force the authors' choice. No circular step can be exhibited from the paper's equations.

Assumptions & free parameters 1 free parameters · 5 assumptions · 3 invented entities

There are no numerical parameters fitted to data; k1,k2,k3 are the given inputs of the family from [24]. The one hand-chosen element is the poset weight scheme of Definition 7.1, which mirrors the exchange polynomials and cancels algebraically. The central claims rest on standard generalized cluster algebra theory [42,45], the published c-vector classification of the Markov cluster algebra [41], and the authors' own prior published work [23,24] for the generalized equation and growth inequalities. The new objects (CGC/CMM matrices and the posets) are fully constructed within the paper and carry checkable external handles, so no invented entity lacks independent evidence.

free parameters (1)
  • poset element weights (k_a * xhat_a and (1/k_a) * xhat_a)
    The Construction Algorithm (Definition 7.1) assigns these weights to paired elements of the fence poset. The scheme is chosen ad hoc to mirror the exchange polynomial Z_a(u) = 1 + k_a u + u^2, and the rational factor 1/k_a cancels in every order-ideal product. This is a construction choice, not a number fitted to external data.
assumptions (5)
  • standard math Generalized cluster algebra theory of Nakanishi: mutation rules, separation formula (Theorem 2.10), and c-vector/g-vector identities (Propositions 2.5, 2.8, and [42, Proposition 3.21]).
    Used throughout Sections 3 and 8 (Theorem 3.21, Lemma 3.25, proof of Theorem 8.30). Cited from [42,45] and treated as established background.
  • domain assumption The c-vector classification of the Markov cluster algebra (Nájera Chávez, Proposition 3.24) extends to the generalized Markov cluster algebras via the diagonal rescaling c_{ij;t} = (d_i/d_j) c^M_{ij;t} of Lemma 3.25.
    Load-bearing input for the F-polynomial recursion used in the induction proving Theorem 8.30 (Section 8.4, equations (8.2) and (8.3)).
  • domain assumption The four seed types (I)-(IV) in Section 3.1 define the generalized Markov cluster algebras A(k1,k2,k3), with exchange polynomials Z_i(u) = 1 + k_i u + u^2 encoding generalized mutation, and all positive integral solutions of equation (1.1) are reachable from (1,1,1) by Vieta jumps.
    Definitions inherited from [24]; the paper's objects are built on these seeds, and the reachability statement is quoted from [24].
  • domain assumption The growth inequality |(a^2 + k_c a b + b^2)/c| > max{|a|,|b|} for generalized Markov clusters.
    Quoted from [23, Proposition 2.4] and [24, Proposition 4]; used in Corollaries 3.10, 3.15 and in the descent arguments of Theorems 4.13 and 5.9.
  • standard math Cayley-Hamilton theorem and the square-root lemma for 2x2 matrices (Lemma 6.11, attributed to [52]).
    Used in Lemma 6.10 to recover Y from Y^2 in the proof of uniqueness of Markov-monodromy decompositions.
invented entities (3)
  • Cluster generalized Cohn (CGC) matrices and CGC triples independent evidence
    purpose: Matrixization of cluster variables in A(k1,k2,k3) with trace condition tr(P) = M p12 - k_{p12}; specializes at x_i=1 to the generalized Cohn matrices of [23].
    Well-defined by Definition 4.1; existence proved by construction (Propositions 4.3 and 4.4). Falsifiable handle: at x1=x2=x3=1 they must reproduce the integer matrices of [23], asserted in Section 11.1.
  • Cluster Markov-monodromy (CMM) matrices and CMM triples independent evidence
    purpose: Alternative matrixization with trace tr(X) = -k_{x12} and product XYZ = T; related to CGC matrices by the isomorphism Psi_g of Theorem 6.4.
    Definition 5.1; existence via Proposition 5.3. Falsifiable handles: the tree isomorphism with CGC trees (Theorem 6.4) and the stated specialization to [23]'s Markov-monodromy matrices.
  • Weighted fence posets P_{p/q}, ~P_{p/q}, P-circle_{p/q}, and the poset H independent evidence
    purpose: Combinatorial model whose order ideals expand cluster variables; weights encode the k_i and the exchange polynomials; the g-vector of P_{p/q} matches the cluster g-vector by direct count (Lemma 7.5).
    Constructions in Section 7. Falsifiable handles: in the k_i=0 case they must reproduce the snake-graph and fence-poset expansions of [26,27,48], and the Christoffel-word interpretation of Remark 7.4.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Cluster algebraic interpretation of generalized Markov numbers and their matrixizations." pith.science (2026). https://pith.science/paper/KPW5RUE3

@misc{pith2026250706900,
  author       = {Pith},
  title        = {Pith review of: Cluster algebraic interpretation of generalized Markov numbers and their matrixizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPW5RUE3}},
  note         = {Machine review of arXiv:2507.06900}
}
abstract

Markov numbers, i.e. positive integers appearing in solutions to $x^2 + y^2 + z^2 = 3xyz$, can be viewed as specializations of cluster variables. The second author and Matsushita gave a generalization of the Markov equation, $x^2 + y^2 + z^2 + k_1yz + k_2xz + k_3xy = (3+k_1+k_2+k_3)xyz$, whose solutions can be viewed as specializations of cluster variables in generalized cluster algebras. We give two families of matrices in $SL(2,\mathbb{Z}[x_1^\pm,x_2^\pm,x_3^\pm])$ associated to these cluster structures. These matrix formulas relate to previous matrices appearing in the context of Markov numbers, including Cohn matrices and generalized Cohn matrices given by the second author, Maruyama, and Sato, as well as matrices appearing in the context of cluster algebras, including matrix formulas given by Kanatarc{\i} O\u{g}uz and Y{\i}ld{\i}r{\i}m. We provide a classification of the two families of matrices and exhibit an explicit family of each. The latter is done by realizing cluster variables in generalized Markov cluster algebras as weight-generating functions of order ideals in certain fence posets which are related to Christoffel words. An interesting observation is that these functions resemble Caldero-Chapoton functions for string modules, and a byproduct of our proofs is a new skein-like formula for such functions.

Figures

Figures reproduced from arXiv: 2507.06900 by the authors.

Figure 1
Figure 1. The poset Pe 1 0 when all ki > 0. Note that if k2 = 0, then this is a chain. generalized Cohn matrices and triples satisfy Definitions 4.1 and 4.2 into a series of inter￾mediate results. Proposition 9.4. For each p q ∈ Q ∩ [0,∞), det(Cp q ) = 1. Proof. Notice the determinant of Cp q is given by x 2g ◦ p q  W(Pe p q ; R)W(Pe p q ; ¬L, ¬R) − W(Pe p q ; ¬R)W(Pe p q ; R, ¬L)  so it suffices to show that W(Pe p q ; R)W… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Monoidal categorification of generalized cluster algebras and conjectures of Fraser and Gleitz

    math.RT 2026-06 accept novelty 7.0 of 10

    For k=3 and any ℓ≥2 the Grothendieck ring of C_{ε,ξ} is isomorphic to a generalized cluster algebra of rank 2ℓ−2, confirming the first half of Gleitz’s conjecture.

  2. Divisibility by $p$ for Markoff-like Surfaces

    math.NT 2025-09 conditional novelty 6.0 of 10

    For most parameters with nonzero 3+a1+a2+a3 and p>=5, every nontrivial orbit on the generalized Markov surface has size divisible by p, with quadratic obstructions giving at least two or four orbits in special cases.

Reference graph

Works this paper leans on

53 extracted references · 48 canonical work pages · cited by 2 Pith papers

  1. [23]

    arXiv:2407.08203

    Yasuaki Gyoda, Shuhei Maruyama, and Yusuke Sato,SL(2,Z)-matrixizations of generalized Markov numbers, 2024. arXiv:2407.08203

    arXiv 2024

  2. [43]

    41, Mathematical Society of Japan, Tokyo, 2023

    , Cluster algebras and scattering diagrams, MSJ Memoirs, vol. 41, Mathematical Society of Japan, Tokyo, 2023. MR4563311

    work page 2023

  3. [1]

    A mathematical journey from irrational numbers to perfect matchings

    Martin Aigner,Markov’s theorem and 100 years of the uniqueness conjecture, Springer, Cham, 2013. A mathematical journey from irrational numbers to perfect matchings. MR3098784

    work page 2013

  4. [2]

    Socially-Aware Robot Navigation Enhanced by Bidirectional Natural Language Conversations Using Large Language Models

    Esther Banaian, Wonwoo Kang, and Elizabeth Kelley,Skein relations for punctured surfaces, 2024. arXiv:2409.049657

    work page Pith review arXiv 2024

  5. [3]

    63 (2024), no

    Esther Banaian and Archan Sen,A generalization of Markov numbers, Ramanujan J. 63 (2024), no. 4, 1021–1055. MR4721155

    work page 2024

  6. [4]

    Algebra674 (2025), 77–116

    Esther Banaian and Yadira Valdivieso,Snake graphs and Caldero-Chapoton functions from triangu- lated orbifolds, J. Algebra674 (2025), 77–116. MR4881531 CLUSTER ALGEBRAIC INTERPRETATION 73

    work page 2025

  7. [5]

    Represent

    Andre Beineke, Thomas Brüstle, and Lutz Hille,Cluster-cyclic quivers with three vertices and the Markov equation, Algebr. Represent. Theory14 (2011), no. 1, 97–112. With an appendix by Otto Kerner. MR2763295

    work page 2011

  8. [6]

    Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky,Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J.126 (2005), no. 1, 1–52. MR2110627

    work page 2005

Show all 53 references
  1. [7]

    arXiv:2503.03719

    Amanda Burcroff, Kyungyong Lee, and Lang Mou,Positivity of generalized cluster scattering dia- grams, 2025. arXiv:2503.03719

    2025 arXiv

  2. [8]

    Michael C. R. Butler and Claus Michael Ringel,Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra15 (1987), no. 1-2, 145–179. MR876976

    1987

  3. [9]

    Algebra 382 (2013), 240–281

    Ílke Çanakçı and Ralf Schiffler,Snake graph calculus and cluster algebras from surfaces, J. Algebra 382 (2013), 240–281. MR3034481

    2013

  4. [10]

    , Snake graphs and continued fractions , European J. Combin. 86 (2020), 103081, 19. MR4058266

    2020

  5. [11]

    GiovanniCerulliIrelli,DanielLabardini-Fragoso,andJanSchröer, Caldero-Chapoton algebras,Trans. Amer. Math. Soc.367 (2015), no. 4, 2787–2822. MR3301882

    2015

  6. [12]

    Leonid Chekhov and Michael Shapiro,Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables, Int. Math. Res. Not. IMRN10 (2014), 2746–2772. MR3214284

    2014

  7. [13]

    arXiv:2501.09435

    Zhichao Chen and Zixu Li,A cluster theory approach from mutation invariants to diophantine equa- tions, 2025. arXiv:2501.09435

    2025 arXiv

  8. [14]

    Harvey Cohn,Approach to Markoff’s minimal forms through modular functions, Ann. of Math. (2) 61 (1955), 1–12. MR67935

    1955

  9. [15]

    18 (1971), 125–136

    , Representation of Markoff’s binary quadratic forms by geodesics on a perforated torus, Acta Arith. 18 (1971), 125–136. MR288079

    1971

  10. [16]

    Anna Felikson, Michael Shapiro, and Pavel Tumarkin,Cluster algebras and triangulated orbifolds, Adv. Math. 231 (2012), no. 5, 2953–3002. MR2970470

    2012

  11. [17]

    15(2002), 497–529

    SergeyFominandAndreiZelevinsky, Cluster algebras I: Foundations,J.Amer.Math.Soc. 15(2002), 497–529. MR1887642

    2002

  12. [18]

    Math.143 (2007), no

    , Cluster algebras IV: Coefficients, Compos. Math.143 (2007), no. 1, 112–164. MR2295199

    2007

  13. [19]

    arXiv:2409.08333

    Sarafina Ford, Amrei Oswald, and James Jian Zhang,Homological conditions on locally gentle alge- bras, 2024. arXiv:2409.08333

    2024 arXiv

  14. [20]

    Robert Fricke, Über die Theorie der automorphen Modulgruppen, Nach. Akad. Wiss. Göttingen (1896), 91–101

  15. [21]

    G Frobenius,Uber die Markoffschen Zahlen, SB Preuss. Akad. Wiss. Berlin (1913) 458–487; avail- able in Gesammelte Abhandlungen Band III, Springer, 1968

    1913

  16. [22]

    Math.290 (2016), 364–452

    Christof Geiß, Daniel Labardini-Fragoso, and Jan Schröer,The representation type of Jacobian al- gebras, Adv. Math.290 (2016), 364–452. MR3451928

    2016

  17. [24]

    Yasuaki Gyoda and Kodai Matsushita,Generalization of Markov Diophantine equation via general- ized cluster algebra, Electron. J. Combin.30 (2023), no. 4, Paper No. 4.10, 20. MR4657283

    2023

  18. [25]

    Reihe, Bd

    Adolf Hurwitz,Über eine Aufgabe der unbestimmten Analysis: Archiv der Mathematik und Physik, III. Reihe, Bd. 11, 1907, S. 185–196, Mathematische Werke: Zweiter Band Zahlentheorie Algebra und Geometrie (1963), 410–421

    1963

  19. [26]

    348 (2025), no

    Ezgi Kantarcı Oğuz,Oriented posets, rank matrices andq-deformed Markov numbers, Discrete Math. 348 (2025), no. 2, Paper No. 114256, 17. MR4797185

    2025

  20. [27]

    MR4863786

    Ezgi Kantarcı Oğuz and Emine Yıldırım,Cluster expansions: T-walks, labeled posets and matrix calculations, Journal of Algebra669 (2025), 183–219. MR4863786

    2025

  21. [28]

    arXiv:2008.12913

    Takeyoshi Kogiso,q-Deformations and t-Deformations of the Markov triples, 2020. arXiv:2008.12913

    2020 arXiv

  22. [29]

    Algebra520 (2019), 90–135

    Daniel Labardini-Fragoso and Diego Velasco,On a family of Caldero-Chapoton algebras that have the Laurent phenomenon, J. Algebra520 (2019), 90–135. MR3880865

    2019

  23. [30]

    Clément Lagisquet, Edita Pelantová, Sébastien Tavenas, and Laurent Vuillon,On the Markov num- bers: fixed numerator, denominator, and sum conjectures, Adv. in Appl. Math.130 (2021), Paper No. 102227, 28. MR4265545

    2021

  24. [31]

    Ludivine Leclere and Sophie Morier-Genoud,q-deformations in the modular group and of the real quadratic irrational numbers, Adv. in Appl. Math.130 (2021), Paper No. 102223, 28. MR4265544

    2021

  25. [32]

    Kyungyong Lee, Li Li, Michelle Rabideau, and Ralf Schiffler,On the ordering of the Markov numbers, Adv. in Appl. Math.143 (2023), Paper No. 102453, 29. MR4505398 74 ESTHER BANAIAN AND YASUAKI GYODA

    2023

  26. [33]

    Differential Geom.48 (1998), no

    Feng Luo,Geodesic length functions and Teichmüller spaces, J. Differential Geom.48 (1998), no. 2, 275–317. MR1630186

    1998

  27. [34]

    Markoff,Sur les formes quadratiques binaires indéfinies, Math

    A. Markoff,Sur les formes quadratiques binaires indéfinies, Math. Ann.17 (1880), no. 3, 379–399. (Sécond mémoire). MR1510073

  28. [35]

    Math.233 (2013), 207–247

    Greg Muller,Locally acyclic cluster algebras, Adv. Math.233 (2013), 207–247. MR2995670

    2013

  29. [36]

    arXIv:2503.21872

    Gregg Musiker,Super Markov Numbers and Signed Double Dimer Covers, 2025. arXIv:2503.21872

    2025 arXiv

  30. [37]

    MR2721552

    Gregg Musiker and Ralf Schiffler,Cluster algebras of unpunctured surfaces and snake graphsAK (2009), 673–684. MR2721552

    2009

  31. [38]

    Gregg Musiker, Ralf Schiffler, and Lauren Williams,Positivity for cluster algebras from surfaces, Adv. Math. 227 (2011), no. 6, 2241–2308. MR2807089

    2011

  32. [39]

    , Bases for cluster algebras from surfaces, Compos. Math. 149 (2013), no. 2, 217–263. MR3020308

    2013

  33. [40]

    Gregg Musiker and Lauren Williams,Matrix formulae and skein relations for cluster algebras from surfaces, Int. Math. Res. Not. IMRN13 (2013), 2891–2944. MR3072996

    2013

  34. [41]

    Alfredo Nájera Chávez,On the c-vectors and g-vectors of the Markov cluster algebra, Sém. Lothar. Combin. 69 (2012), Art. B69d, 12. MR3118908

    2012

  35. [42]

    Math.277 (2015), no

    Tomoki Nakanishi,Structure of seeds in generalized cluster algebras, Pacific J. Math.277 (2015), no. 1, 201–217. MR3393688

    2015

  36. [44]

    structure of seeds in generalized cluster algebras

    , Addendum to “structure of seeds in generalized cluster algebras", 2024. arXiv:2406.07582

    2024 arXiv

  37. [45]

    MR3643935

    Tomoki Nakanishi and Dylan Rupel,Companion cluster algebras to a generalized cluster algebra24 (2016), 129–149. MR3643935

    2016

  38. [46]

    Tomoki Nakanishi and Andrei Zelevinsky,On tropical dualities in cluster algebras565 (2012), 217–

    2012

  39. [47]

    Toshihiro Nakanishi and Marjatta Näätänen,Areas of two-dimensional moduli spaces, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3241–3252. MR1844999

    2001

  40. [48]

    arXiv:2311.06033

    Vincent Pilaud, Nathan Reading, and Sibylle Schroll,Posets for F-polynomials in cluster algebras from surfaces, 2023. arXiv:2311.06033

    2023 arXiv

  41. [49]

    MR4067101

    James Propp, The combinatorics of frieze patterns and Markoff numbers, INTEGERS 20 (2020), A12. MR4067101

    2020

  42. [50]

    Michelle Rabideau and Ralf Schiffler,Continued fractions and orderings on the Markov numbers, Adv. Math. 370 (2020), 107231, 18. MR4103773

    2020

  43. [51]

    Robert Remak, Über indefinite binäre quadratische minimalformen, Math. Ann. 92 (1924), no. 3, 155–182. MR1512210

    1924

  44. [52]

    Donald Sullivan, The square roots of 2× 2 matrices, Math. Mag. 66 (1993), no. 5, 314–316. MR1251447

    1993

  45. [53]

    arXiv:math/0606283

    Ying Zhang,An elementary proof of uniqueness of Markoff numbers which are prime powers, 2006. arXiv:math/0606283. (EstherBanaian) Department of Mathematics, University of California, Riverside, River- side, CA, 92501, United States Email address: estherb@ucr.edu (YasuakiGyoda)...

    2006 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.