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On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges

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Pith's one-line read The paper proves every Markov triple has a unique q-analogue, tied to weighted snake-graph matchings.

desk verdict Solid q-analogue paper with a real but fixable mismatch between the abstract's uniqueness claim and the theorem actually proved. read the letter →

arxiv 2507.19080 v1 pith:B5NBUCRX submitted 2025-07-25 math.CO

Sam Evans , Perrine Jouteur , Sophie Morier-Genoud , Valentin Ovsienko This is my paper · ORCID
classification math.CO MSC 05A3005C7011A5511J70
keywords q-MarkovnumbersMarkovequationLaurentpolynomialsCohnmatricessnakegraphsperfectmatchingsq-deformedrationalsmodulargroup
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 introduces a q-analogue of Markov numbers: Laurent polynomials that specialise to classical Markov numbers at $q=1$ and satisfy a q-deformed Markov equation. Its central theorem says every Markov triple has exactly one such q-deformation, computable from the trace of any q-deformed Cohn matrix. The main result identifies each q-Markov number with the weighted count of perfect matchings of a snake graph whose boundary edges carry alternating weights $q$ and $q^{-1}$. These polynomials are monic, palindromic, non-negative, and unimodal, and the rational label of a Markov number can be recovered from the top two coefficients. The construction gives a graded, combinatorial refinement of the classical Markov tree.

What carries the argument

The machinery has three linked components. First, the q-Markov equation with the Vieta-like mutation $c'_q = q^{-1}[3]_q a_q b_q - c_q$, which generates every q-Markov triple from $(1,1,1)$. Second, the q-deformed Cohn matrices $A(n)_q$, $B(n)_q$ obtained from the q-deformed modular group generators $T_q$, $S_q$, $L_q$; their traces, divided by $q^{-1}[3]_q$, produce the q-Markov numbers and are independent of the integer parameter $n$. Third, the weighted snake graph $G_t(q)$, whose edges on the western and southern borders alternate $q^{-1}, q, \dots$ and whose other edges have weight $1$; the weighted matching count $\mu_t(q)$ reproduces $m_t^q$, and the proof identifies the recurrence for matchings with multiplication by the same q-deformed Cohn matrices.

What would settle it

Take two distinct rational labels $t,t'$ in $[0,1]$ with the same classical Markov number (none are known, but the conjecture is open), compute $m_t^q$ and $m_{t'}^q$ from the recurrence; if the Laurent polynomials differ, uniqueness of the q-deformation for that integer fails. Even without such a collision, one can recompute the weighted matching polynomial $\mu_t(q)$ for a label such as $t=3/5$ by direct enumeration of perfect matchings and compare it coefficient-by-coefficient with the recurrence: any mismatch would falsify Theorem 16.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Markov equation admits a canonical q-analogue: for every Markov triple $(a,b,c)$ there exists a unique triple of Laurent polynomials $(a_q,b_q,c_q)$ satisfying the q-Markov equation and reducing to $(a,b,c)$ at $q=1$. The q-Markov number $m_t^q$ can be read off as $\mathrm{Tr}(C_t^q)/(q^{-1}[3]_q)$ for any q-deformed Cohn matrix in the tree, independent of the choice of initial matrices; equivalently, it is the generating polynomial $\mu_t(q)$ that counts perfect matchings of the weighted snake graph $G_t(q)$. This gives a graded lift of the classical fact that Markov numbers count perfect matchings, and it implies that each q-Markov number determines its rational label $t$ via an explicit degree-coefficient formula.

Load-bearing premise

The proof that every Markov number, as opposed to every Markov triple, has a unique q-analogue assumes the classical Markov uniqueness conjecture, which is still open; the proved statement is uniqueness for each labelled triple.

Editorial extensions

If this is right

  • Every Markov triple has a canonical q-analogue, so all classical Markov numbers carry a canonical Laurent-polynomial refinement when the classical uniqueness conjecture holds.
  • The q-Markov numbers are monic palindromic Laurent polynomials with non-negative integer coefficients, and all but $2_q$ have unimodal coefficient sequences.
  • The weighted perfect-matching model gives a positive combinatorial formula for $m_t^q$, so each coefficient counts matchings with a fixed weight.
  • The map $t \mapsto m_t^q$ from rationals in $[0,1]$ to Laurent polynomials is injective, and $t$ is explicitly recovered from the degree and the next coefficient.
  • Trace computations with q-deformed Cohn matrices are independent of the chosen initial matrices, even though the upper-right entries are not.

Reading between the lines

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

  • If the classical uniqueness conjecture fails, the stated uniqueness of each integer Markov number's q-analogue becomes ambiguous: the theorem proves uniqueness per labelled triple, not per integer, so two labels with the same integer could in principle yield different q-polynomials.
  • The weighted snake-graph model suggests a dimer-theoretic reading in which coefficients of $m_t^q$ are refined matching counts; the same weighting rule could plausibly be tested on Christoffel words outside the range $0\le t\le 1$ to produce new q-rational identities.
  • Because the recurrence and matching proof are constructive, the same technique may extend to q-deformations of other cluster-algebra mutations, such as generalized Markov equations, where a similar weighted snake graph could be built.
  • The formula $t=(d-\alpha)/(\alpha+1)$ gives a finite, direct test for injectivity on any finite set of rational labels, so numerical computation of the recurrence could probe the analogue of the Markov injectivity conjecture in this graded setting.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper introduces a q-deformation of Markov numbers as Laurent polynomial solutions to the q-Markov equation (2), proves that each Markov triple has a unique such deformation, identifies the resulting q-Markov numbers with traces of q-deformed Cohn matrices independently of the chosen initial matrices, and gives a weighted snake graph model in which q-Markov numbers are generating functions of perfect matchings. It also proves positivity, palindromicity, unimodality, and injectivity of the labeled map t ↦ m_t^q. The central technical tools are a mutation descent (Proposition 2), a transfer-matrix computation (Theorem 16), and an induction on the Cohn tree (Theorem 12 and Lemma 13).

Significance. The paper gives a clean framework that unifies earlier ad hoc definitions of q-Markov numbers and provides a new combinatorial interpretation via dimer coverings with edge weights. The trace-invariance result is a conceptual improvement over definitions depending on a choice of Cohn matrices. The proofs are largely constructive, the statements are concrete and checkable, and the paper ships detailed worked examples. The main caveat is that the advertised number-level uniqueness is conditional on the classical (open) Markov uniqueness conjecture; the body states this, but the abstract does not.

major comments (2)
  1. [Abstract and Section 1] The statement 'every Markov number has a unique q-deformation' is not proved in the manuscript. Theorem 1 proves uniqueness for ordered Markov triples, and the body (Section 1, near the end of the introduction) explicitly says that the passage from triples to numbers uses 'the assumption that the Markov conjecture holds true'. Corollary 4 only shows that the labeled polynomials m_t^q are distinct for t ≠ t′; if the classical conjecture were false and two labels t,t′ had m_t = m_t′, the same integer would have two different q-polynomials. The abstract and the introductory paragraph should be reworded so that number-level uniqueness is stated as conditional on the Markov uniqueness conjecture, or replaced by the proved triple-level and labeled statements.
  2. [Section 3.1, Proposition 2] The degree-descent proof is incomplete when d1 = -1. From d3 = d1 + d2 + 1 and d1 ≤ d2 ≤ d3, the value d1 = -1 forces d2 = d3, in which case the coefficient of q^{2d3} on the left-hand side of (2) is α2^2 + α3^2, not α3^2, and the displayed relation α3^2 = α1α2α3 does not hold. The manuscript does not address this case. The case can be ruled out by comparing the coefficient of q^{2d3-1}, which gives a contradiction α1 = 0, so the statement of Proposition 2 is likely correct, but the proof as written needs this additional argument or an explicit justification that all degrees are nonnegative. This is load-bearing because Proposition 2 underpins Theorem 1.
minor comments (6)
  1. [Figure 1 caption] The word 'weigth' should be 'weight'.
  2. [Definition 15(ii)] The phrase 'the the weighted number' contains a duplicated article.
  3. [Section 4.3, proof of Theorem 12] The identity A(n)_q B^{-1}(n)_q = A(n-2)_q is stated without proof; a one-line verification would improve readability.
  4. [Section 5.3, proof of Theorem 16] The palindrome property of w1...ws used to reverse the order of the matrices M_i is not proved or referenced; it follows from standard Christoffel word theory and should be stated explicitly.
  5. [Section 5.5, Lemma 17] The uniqueness of the minimal perfect matching and the count α = #X's - 1 are asserted without a full justification; a short argument or a precise reference would make the proof of Corollary 4 more self-contained.
  6. [Section 6.3] The claim that specializing y1 = y2 = y3 = q in \hat A, \hat B returns A(2)_q, B(2)_q 'up to a power of q' is imprecise; please specify the normalization explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: q-Markov numbers are defined by equation (2), with Cohn-trace and snake-graph matches proved independently.

full rationale

The paper's derivation chain is self-contained and non-circular. q-Markov triples are introduced through equation (2), and Theorem 1 is proved by the degree-descent recurrence of Proposition 2 together with the classical uniqueness of Markov triples in the Markov tree (Aigner, Thm 3.3). The Cohn-matrix result (Theorem 12) is established by a direct induction from the explicitly written q-deformed matrices A(n)_q and B(n)_q, using the trace identity Tr(MN)=Tr(M)Tr(N)-Tr(MN^{-1}) and the Vieta transform; it does not presuppose the trace formula. The snake-graph theorem (Theorem 16) independently defines a weighted matching generating function µ_t(q), derives the recurrences involving A(1)_q and B(1)_q from the graph structure, and then identifies µ_t(q) with the trace via Lemma 13. Citations to [22,25] provide the q-deformed modular-group context and terminology, but the matrices and relations are stated and verified directly in the paper, so the self-citations are not load-bearing. The only notable issue is that the abstract's unconditional claim that every Markov number has a unique q-deformation goes beyond what is proved: Section 1 explicitly says this conclusion holds only "under the assumption that the Markov conjecture holds true", since Theorem 1 concerns ordered triples rather than numbers that may occur in multiple triples. This is a conditionality/correctness concern about the abstract, not a circular reduction of the derivation itself.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data; the weight assignment on snake graph edges is a fixed rule, not a parameter optimized to match outputs. The q-variable is a formal symbol. The paper relies on standard prior results: Markov's mutation theorem, Aigner's classification of Cohn matrices, the snake-graph perfect matching theorem, and the q-deformed PSL(2,Z) matrices of [22]. The only open-conjecture input is the classical Markov uniqueness conjecture, needed for the unqualified statement in the abstract that every Markov number has a unique q-deformation.

assumptions (7)
  • standard math Markov's theorem: every Markov triple is obtained from (1,1,1) by repeated mutations (a,b,c) -> (a,b,3ab-c) and permutations.
    Used in Proposition 2 and Theorem 1 to characterize all q-Markov triples via the Vieta recurrence; cited to [1, p.46].
  • standard math Aigner's classification of Cohn matrices: every Cohn matrix is equal, up to the tree structure, to A(n) or B(n) for some integer n.
    Basis for defining q-deformed Cohn matrices A(n)_q and B(n)_q in Section 4.2 and for the trace independence proof in Theorem 12; cited [1, Thm 4.8].
  • standard math The classical Markov numbers count perfect matchings of snake graphs (Propp's theorem).
    Motivates and anchors the weighted snake graph model in Section 5; the paper effectively re-proves a weighted version via transfer matrices.
  • domain assumption The matrices T_q, S_q, L_q satisfy the PSL(2,Z) relations and generate the q-deformed modular group action from [22].
    Used to construct the q-deformed Cohn matrices in Section 4.1; this is prior work by two of the authors, stated in [22] and [25].
  • domain assumption The classical Markov uniqueness conjecture (Frobenius) is assumed when claiming every integer Markov number has a unique q-deformation.
    The body explicitly conditions this statement on the conjecture; the abstract omits the condition. The proved results (Theorem 1, Corollary 4) do not require the conjecture for labeled triples.
  • standard math Unimodality of rank polynomials of fence posets [31] and their equality with trace-normalized q-Markov polynomials [30].
    Used in Proposition 6 to conclude that q-Markov coefficients are unimodal except for 2_q; both results are cited, not proved.
  • standard math The middle segment of the A/B Christoffel word is a palindrome.
    Used in the proof of Theorem 16 to reverse a product of transfer matrices; this is a standard property of Christoffel words.

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Pith. "Pith review of On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges." pith.science (2026). https://pith.science/paper/B5NBUCRX

@misc{pith2026250719080,
  author       = {Pith},
  title        = {Pith review of: On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B5NBUCRX}},
  note         = {Machine review of arXiv:2507.19080}
}
abstract

We consider a natural $q$-deformation of the classical Markov numbers. This $q$-deformation is closely related to $q$-deformed rational numbers recently introduced by two of us. Both notions, those of $q$-rationals and $q$-Markov numbers, are based on invariance with respect to the action of the modular group $mathrm{PSL}(2,\mathbb{Z})$. We prove that every Markov number has a unique $q$-deformation, which is a monic unimodal palindromic Laurent polynomial with positive integer coefficients. The $q$-Markov numbers can be calculated in terms of the traces of $q$-deformed Cohn matrices, and we show that $q$-Markov numbers are independent of the choice of such matrices. We construct a combinatorial model counting perfect matchings of snake graphs with weighted edges.

Figures

Figures reproduced from arXiv: 2507.19080 by the authors.

Figure 1
Figure 1. A snake graph with weighted edges. Edges coloured in red have weight [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Markov numbers represented on the Conway topograph. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The fishbone The tree of Markov numbers can be compared with the yet more ancient and better known Farey tree; see, e.g. [16, 1, 34]. 1 1 0 1 1 2 2 3 3 5 3 4 · · · 5 8 · · · · · · 1 3 2 5 1 4 · · · · · · 2 7 1 5 1 0 · · · [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The Farey tree of rational numbers. Rational numbers label regions bounded by the binary tree, and obey the following local rule: every 5 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: The Farey sum. It is known that the Farey tree contains all rational numbers Q ∪ 1 0 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: The Cohn tree. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Snake graphs. The Cohn snake (left) vs the domino snake Gt (right). The construction of the snake graph that will be relevant for us refines that of Cohn. Tracing the “median zig-zag line” joining the point ( 1 2 , 0) and (s, k − 1 2 ) (see the red line on [PITH_FULL_…
Figure 8
Figure 8. Figure 8: Examples of perfect matchings. There are two perfect matchings for a single box and three for [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: The tree of Christoffel words in X, Y Let us collect here some important facts about Christoffel words. 1. Every Christoffel word is of the form w = XπY where π is a palindromic word in X, Y ; see [34, Thm 2.3.1]. 2. The rational number corresponding to wt is given by …
Figure 10
Figure 10. Figure 10: Cohn branchings. and we know that Tr(M) = q −1 [3]q mt , Tr(N) = q −1 [3]q mt ′ , Tr(MN −1 ) = q −1 [3]q mt⊖t ′ , (13) where t ⊖ t ′ is the operation opposite to the Farey sum t ⊕ t ′ : r s ⊖ r ′ s ′ := r − r ′ s − s ′ . We need to prove that Tr(MN) = q −1 [3]q mt⊕t ′…
Figure 11
Figure 11. Figure 11: Markov numbers on the Conway topograph, in all directions. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Cohn tree in all directions. The tree is oriented, at every vertex of the tree there are exactly one ingoing and two outgoing edges. Without this condition, the construction is not consistent. The rule for constructing the tree and calcu￾lating the matrices in it is a…
Figure 13
Figure 13. Figure 13: Weighted graph G˜ t(q) for t = 2 3 . More generally, in the q-deformed matrices of PSL(2, Z) the top right entry corresponds to a numerator of a q-rational [25]. The above observation leads to another combinatorial model for the q-rationals. The details will be develo…
Figure 14
Figure 14. Figure 14: Branching rule for cluster variables (=mutation) [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: Tree of cluster variables In a more general setting [13], cluster variables are elements of Z>0[y1, y2, y3][x ±1 1 , x±1 2 , x±1 3 ]. In this setting, one recovers the cluster variables Xt described above by specializing y1 = y2 = y3 = 1, and one obtains the associate…
Figure 16
Figure 16. Figure 16: Tree of F-polynomials. Example 5. (a) Further examples of Markov cluster variables are as follows. X1 3 = x 6 1 + 3x 4 1x 2 2 + 3x 2 1x 4 2 + x 6 2 + (2x 4 1 + 2x 2 1x 2 2 )x 2 3 + x 2 1x 4 3 x 2 2x 3 3 , X1 4 = (x 8 1 + 4x 6 1x 2 2 + 6x 4 1x 4 2 + 4x 2 1x 6 2 + x 8 2…

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