Nuancing the unicity of $q$-rationals

Alexander Thomas; Olga Paris-Romaskevich; Perrine Jouteur

REVIEW 2 major objections 3 minor 15 references

Nuancing the unicity of $q$-rationals

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read There are exactly two equivariant deformations of rationals that extend the classical q-integers.

desk verdict A genuinely new t=0 deformation and a plausible 'exactly two' theorem, but the completeness proof in Theorem 3.1 has a gap that needs closing before the unicity claim is fully supported. read the letter →

arxiv 2607.22308 v1 pith:MU3F6SYA submitted 2026-07-24 math.QA math.COmath.GT

Perrine Jouteur , Olga Paris-Romaskevich , Alexander Thomas This is my paper · ORCID

classification math.QAmath.COmath.GT MSC 05A3011F0611A55

keywords q-rationalsq-integersmodulargroupcharactervarietydeformedrationalnumberspositivityJonespolynomialcontinuedfractions

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 asks how many ways there are to extend the q-integers [n]_q = (1-q^n)/(1-q) to all rational numbers while keeping a modular-group symmetry. It proves, via the SL2(C) character variety of the modular group, that every non-abelian representation of PSL2(Z) in PSL2(C) is conjugate to the standard two-generator representation, so q-rationals are unique up to conjugacy. Yet after fixing the generator T to preserve the q-integers, a two-parameter family of deformations appears, and exactly two members of that family—the classical right q-rationals and a new left version—actually deliver the usual q-integers on integers. The new member retains positivity and yields a direct computation of the Jones polynomial of rational knots.

What carries the argument

The SL2(C) character variety of the free group on two generators, coordinatized by three traces (x,y,z) = (tr g, tr h, tr gh), collapses the modular-group relations to a single parameter q = tr T_q. Solving the equations S^2 = (T_q S)^3 = 1 with T_q fixed produces the one-parameter family S_{q,t}; the special values t=1 and t=0 are picked out by the requirement that [n]_q arise as the orbit of the point at infinity.

What would settle it

Evaluate the discarded root at q=2, compute the candidate matrix for S, and verify whether S^2 = (T_q S)^3 = 1 in PGL2(C) and whether the orbit T_q^n S(∞) produces [n]_q for all n. If it does, the 'exactly two' corollary is incomplete.

Watch

Extended reading notes

Core claim

The central discovery is a complete classification, up to conjugacy, of modular-group equivariant deformations of the projective line that agree with the q-integers on integers. Any non-abelian representation ϱ: PSL2(Z) → PSL2(C) is conjugate to the two-generator representation T_q = [[q,1],[0,1]], S_q = [[0,-1],[q,0]] for some q ∈ C; this is Theorem 2.9. Once T is fixed to T_q, the second generator S can be deformed in a one-parameter family S_{q,t}, and the condition that the orbit of the point at infinity reproduces [n]_q for all n leaves exactly two possibilities: t=1, the original right q-version, and t=0, a new left q-version. The t=0 deformation is conjugate to the standard one by a h

Load-bearing premise

The classification in Theorem 3.1 relies on discarding one root of a quadratic for a matrix entry because it 'does not give the correct value for q=1', without proving that no valid representation for q≠1 survives from that root.

Editorial extensions

If this is right

No third equivariant deformation can match the classical q-integers; the classification is closed unless a discarded branch comes back at special q.
The new t=0 deformation gives a direct combinatorial computation of the Jones polynomial for rational knots, with a formula expressing each (q,0)-deformation as a homothety applied to the original (q,1)-deformation.
The two versions have complementary positivity: at t=0, left numerators and denominators are ordinary polynomials with nonnegative integer coefficients, while right denominators gain an extra factor (q^2 - q + 1).
Deformed Farey determinants stay positive for all four allowed left/right pairings at t=0, extending the positivity theory of Farey-type determinants.
The unicity theorem implies palindromicity of traces and the q-to-q^{-1} conjugacy are intrinsic consequences of the character variety, not extra assumptions.

Reading between the lines

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

The same trace-coordinate method could classify deformations equivariant with respect to Hecke or Coxeter groups; if their character variety is also one-dimensional after fixing the elliptic generator, an analogous two-way dichotomy may appear.
The discarded quadratic root in the proof of Theorem 3.1 is the only place where the 'exactly two' claim could fail; probing it at roots of unity with a direct computation would either close the gap or reveal a third branch.
Because the t=0 deformation is conjugate to the standard one by a homothety depending rationally on q, other rational functions f(q) might generate further positive deformations, potentially linking to other knot invariants.
The direct Jones polynomial formula suggests a poset-theoretic interpretation of the t=0 numerator as a rank-generating function for a modified fence poset, which might extend to more general links.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read. The paper's main positive result — that there are exactly two equivariant deformations of the q-integers, one of which is new — is probably true, but the proof of the 'exactly two' statement is not complete. That doesn't stop me from wanting the paper to be taken seriously.

What's genuinely new: the two-parameter family (3.1), the observation that t=1 and t=0 are the only values giving the usual q-integers, and the t=0 representative with its positivity and fence-poset interpretations. The t=0 version also gives a direct computation of Jones polynomials of rational knots. The character-variety unicity (Theorem 2.9) is a repackaging of known results, but it's done cleanly.

The soft spot is Theorem 3.1, the classification that supports Corollary 3.8. The proof solves for S with T fixed and throws away one root of the quadratic because 'it does not give the correct value for q=1'. That's not a valid reason. For generic q, that root is a perfectly good solution of the PSL_2(C) relations, just with c in a quadratic extension of Q(q). The stress-test's calculation shows it does not send the relevant fixed points to zero, so it probably doesn't produce a third q-integer deformation — but the paper doesn't show that. A referee should ask for a complete argument: either state the coefficient ring and explain why the root is outside it, or handle the algebraic branch and check the fixed-point condition. This is a fixable gap, not a fatal one.

A second issue: the positivity theorem (4.1) leans on Proposition 4.5 from the authors' own companion preprint [6]. That's a same-author dependency that hasn't been independently checked. It's not circular, but it makes the paper's main positivity claim not self-contained.

I don't share the reader's concern about Proposition 2.8. For even n, any lift of an order-n projective element must have n-th power −I, otherwise the projective order would be n/2 or less. And the paper only uses n=2, so the proposition is not a load-bearing issue.

Bottom line: the paper is for people working on q-deformed rationals and rational knot invariants. The new t=0 deformation is interesting, and the 'exactly two' result is worth a careful referee. Send it to peer review. The referee should insist on a fixed completeness proof for Theorem 3.1 and clarity about the reliance on [6].

Referee Report

2 major / 3 minor

Summary. The paper studies q-deformations of rational numbers via representations of the modular group PSL2(Z) into PGL2(C). It proves a unicity theorem (Theorem 2.9) asserting that every non-abelian representation is conjugate to the standard family (1.2) for some q. It then fixes ϱ(T)=T_q and classifies all possible ϱ(S), obtaining a two-parameter family ϱ_{q,t} (Theorem 3.1), and shows that exactly two choices of t deliver the usual q-integers: t=1 (the known right version) and t=0 (a new left version). The paper further studies the t=0 family, proving positivity of numerators/denominators and a direct computation of the Jones polynomial for rational knots.

Significance. If the completeness of the classification is properly established, this is a valuable contribution to the theory of q-deformed rational numbers. The character-variety viewpoint gives a clean conceptual proof of unicity, and the explicit two-parameter family, with its new left version and positivity properties, is an interesting addition. The connection to Jones polynomials of rational knots is a nice application. However, the proof of the central 'exactly two' theorem has a gap in ruling out a branch of solutions to the group relations, which must be repaired before the result can be fully accepted.

major comments (2)

[Section 3, Theorem 3.1 and Corollary 3.8] The proof discards the root b=(q-1)/q of the quadratic with the phrase 'does not give the correct value for q=1'. Since Corollary 3.8's completeness rests on Theorem 3.1, this is not sufficient. One must show that no branch of this root for q≠1 satisfies the q-integer conditions S(∞)=0 or S(1/(1-q))=0. A direct calculation shows that on this branch S(∞)=(q-1)/c and the numerator of S(1/(1-q)) is -1/q, so neither fixed point maps to 0 for generic q; thus the branch is harmless. This check is absent, leaving the 'exactly two' assertion resting on an unproven completeness claim.
[Section 3, proof of Corollary 3.8] The statement 'S_{q,t}[∞]^♭_q = 0 iff t=0' is only valid for constant t. The equation in t also has the q-dependent solution t=-q/(q-1)^2; substituting this into (3.1) yields a matrix with determinant 0, hence not an element of PGL. The proof does not mention that t is a fixed parameter independent of q, nor does it exclude this singular solution. The argument should be made explicit.

minor comments (3)

[Section 3, proof of Theorem 3.1] The assertion 'If a=0, then S_q is the only solution' is stated without proof. A short calculation would make the classification self-contained.
[Section 2.2, proof of Theorem 2.9] The use of Proposition 2.8 to rule out \tilde S^2=1 is terse. Expanding this argument would improve readability.
[Example 3.4 / notation] In Example 3.4, the symbol [n]^#_{q,t} appears instead of [n]^♯_{q,t}; also, the notation PSL_{2,q}(Z) in Corollary 2.13 is not defined. These are minor typographical issues.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the two-parameter family and 'exactly two' corollary are obtained by solving the defining group relations, not by assuming the target. Two caveats (companion-paper dependency and a branch-completeness gap) are correctness/self-containedness issues, not circularity.

full rationale

The main derivation is not circular. Theorem 2.9 classifies non-abelian PSL2(Z)-representations by standard character-variety trace coordinates, with the trace of T as the only free parameter; it does not assume Theorem A. Theorem 3.1 then fixes rho(T)=T_q and solves the two group relations S^2=(T_q S)^3=1 directly for the entries of rho(S); the two-parameter family (3.1) is the solution set of those equations, not a target result inserted by hand. Corollary 3.8 checks which solutions send one of the two T_q-fixed points to 0, which is exactly the condition that the orbit of T_q produces the usual q-integers; the computation t=1 (right) and t=0 (left) is a check, not an assumption. The new-positivity part is less self-contained: Theorem 4.3 is proved by explicitly reducing (q,0)-Farey determinants to (q,1)-determinants via formula (4.1) and then invoking Proposition 4.5 from the same authors' companion preprint [6]. That is a same-author dependency and a self-containedness gap, but not circular: [6] concerns the standard (q,1)-family, not the (q,0) theorem being proved, and the reduction itself is carried out in the present paper. Finally, Theorem 3.1's proof discards the root b=(q-1)/q with the phrase 'does not give the correct value for q=1' without ruling out a valid representation on q!=1 branches; this is a completeness/rigor gap in the 'exactly two' proof, not a circular reduction. These caveats do not make the main derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted constants; q and t are the deformation variables under study. The main unverified import is the companion preprint [6] for (q,1) positivity; the rest is standard character variety background.

assumptions (5)

domain assumption GIT quotient identification: two conjugacy classes are identified when their closures intersect
Used throughout Section 2 to pass from conjugacy classes to points of the character variety; makes Theorem 2.9 an equivalence of semisimple parts rather than literal conjugacy.
standard math χ(F2, SL2(C)) ≅ C3 via traces (x,y,z) = (tr g, tr h, tr gh)
Invoked as Proposition 2.2/Corollary 2.6 (Goldman, Culler–Shalen) to describe representations by trace coordinates in Theorem 2.9.
standard math The formal character variety χ(F2, GL2(Q(q))) ≅ Q(q)3
Cited from Saito [12]; used in Remark 2.7 to justify extending trace-coordinate arguments to q-rational functions.
standard math An order-2 element of PSL2(C) has any lift with square −1
The n=2 case of Proposition 2.8, used in Theorem 2.9 to force tr(S̃)=0.
domain assumption For a/b > c/d, the (q,1)-deformed Farey determinants are in N[q]; and B♯1(σ)+(σ−1)A♯1(σ) ∈ U6, B♭1(σ)+(σ−1)A♭1(σ)=0
Lemma 4.4 and Proposition 4.5 of the companion preprint [6] by the same authors; this is the load-bearing input for the new positivity Theorem 4.3.

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Pith. "Pith review of Nuancing the unicity of $q$-rationals." pith.science (2026). https://pith.science/paper/MU3F6SYA

@misc{pith2026260722308,
  author       = {Pith},
  title        = {Pith review of: Nuancing the unicity of $q$-rationals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MU3F6SYA}},
  note         = {Machine review of arXiv:2607.22308}
}

abstract

We prove unicity of $q$-rational numbers up to conjugacy, using character varieties. Despite the unicity, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual $q$-integers: the original $q$-rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.

Figures

Figures reproduced from arXiv: 2607.22308 by the authors.

**Figure 4.1.** Knot 5 2 from Rolfsen knot table in the Knot Atlas [11] [PITH_FULL_IMAGE:figures/full_fig_p013_4_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 7 linked inside Pith

[6]

Jouteur, O

P. Jouteur, O. Paris-Romaskevich, and A. Thomas. Plane geometry ofq-rationals and Springborn operations. preprint, arXiv/2603.04295, 2026

arXiv 2026
[1]

Bapat, L

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[2]

Culler and P

M. Culler and P. B. Shalen. Varieties of group representations and splittings of 3-manifolds.Ann. Math. (2), 117:109–146, 1983. marc-culler.info/static/home/papers/CharacterVarieties.pdf

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W. M. Goldman. The modular group action on real SL(2)-characters of a one-holed torus.Geom. Topol., 7:443–486, 2003. Eudml/123528

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W. M. Goldman. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. InHand- book of Teichm¨ uller theory. Volume II, pages 611–684. Z¨ urich: European Mathematical Society (EMS), 2009. arXiv/0901.1404

arXiv 2009
[5]

P. Jouteur. Symmetries of theq-deformed real projective line. Preprint, arXiv/2503.02122, 2025

arXiv 2025
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L. H. Kauffman and S. Lambropoulou. On the classification of rational tangles.Advances in Applied Mathematics, 33(2):199–237, 2004. arXiv/0311499

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Leclere and S

L. Leclere and S. Morier-Genoud.q-deformations in the modular group and of the real quadratic irrational numbers.Advances in Applied Mathematics, 130:102223, 2021. arXiv/2101.02953

arXiv 2021

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S. Morier-Genoud and V. Ovsienko.q-deformed rationals andq-continued fractions.Forum Math. Sigma, 8:55, 2020. arXiv/1812.00170

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S. Morier-Genoud and V. Ovsienko. Onq-deformed real numbers.Exp. Math., 31(2):652–660,
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D. Rolfsen. Knot table. katlas.org/wiki/5 2
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K. Saito. Character variety of representations of a finitely generated group in SL 2.Topology and Teichm¨ uller Spaces, pages 253–264, 1996. kurims.kyoto-u.ac.jp/ saito/kojima.pdf

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [6]

Jouteur, O

P. Jouteur, O. Paris-Romaskevich, and A. Thomas. Plane geometry ofq-rationals and Springborn operations. preprint, arXiv/2603.04295, 2026

arXiv 2026

[2] [1]

Bapat, L

A. Bapat, L. Becker, and A. M. Licata.q-deformed rational numbers and the 2-Calabi-Yau cate- gory of typea 2.Forum Math. Sigma, 11:41, 2023. arXiv/2202.07613

arXiv 2023

[3] [2]

Culler and P

M. Culler and P. B. Shalen. Varieties of group representations and splittings of 3-manifolds.Ann. Math. (2), 117:109–146, 1983. marc-culler.info/static/home/papers/CharacterVarieties.pdf

1983

[4] [3]

W. M. Goldman. The modular group action on real SL(2)-characters of a one-holed torus.Geom. Topol., 7:443–486, 2003. Eudml/123528

2003

[5] [4]

W. M. Goldman. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. InHand- book of Teichm¨ uller theory. Volume II, pages 611–684. Z¨ urich: European Mathematical Society (EMS), 2009. arXiv/0901.1404

arXiv 2009

[6] [5]

P. Jouteur. Symmetries of theq-deformed real projective line. Preprint, arXiv/2503.02122, 2025

arXiv 2025

[7] [7]

L. H. Kauffman and S. Lambropoulou. On the classification of rational tangles.Advances in Applied Mathematics, 33(2):199–237, 2004. arXiv/0311499

arXiv 2004

[8] [8]

Leclere and S

L. Leclere and S. Morier-Genoud.q-deformations in the modular group and of the real quadratic irrational numbers.Advances in Applied Mathematics, 130:102223, 2021. arXiv/2101.02953

arXiv 2021