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arxiv: 2604.17401 · v1 · submitted 2026-04-19 · 🧮 math.NT

Markov fractions and Cohn matrices

A.P. Veselov This is my paper

Pith reviewed 2026-05-10 06:01 UTC · model grok-4.3

classification 🧮 math.NT
keywords Markov fractionsCohn matricesConway topographcontinued fractionsMarkov numbersDiophantine approximation
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The pith

Markov fractions coincide with the indices of Cohn matrices, yielding a concatenation rule for their continued fractions.

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

The paper proves that Markov fractions, a recent construction labeling certain rationals tied to Markov numbers, are identical to the indices of Cohn matrices from an earlier matrix formulation. The proof proceeds by comparing their representations as continued fractions and showing they occupy the same positions on the Conway topograph. A reader would care because the match transfers properties between the two setups and replaces separate descriptions with one unified object. The immediate payoff is an explicit rule for building longer continued fractions by concatenation on the topograph.

Core claim

We show that the Markov fractions introduced recently by Springborn coincide with the index of the Cohn matrices defined by Aigner. This provides a simple concatenation rule for the corresponding continued fractions on the Conway topograph.

What carries the argument

The Conway topograph, which arranges continued fractions so that the identified Markov fractions and Cohn-matrix indices occupy matching locations and admit direct concatenation.

Load-bearing premise

The given definitions of Markov fractions and Cohn matrices are compatible and the ordinary rules for continued fractions on the topograph suffice to establish the match.

What would settle it

A single explicit rational that qualifies as a Markov fraction yet fails to equal the index of its associated Cohn matrix when both are computed from the same Markov triple.

Figures

Figures reproduced from arXiv: 2604.17401 by A.P. Veselov.

Figure 1
Figure 1. Figure 1: Conway topograph of rationals in [0, 1]. To present in a similar way all the corresponding Markov fractions one should replace here the Farey mediant by the Springborn mediant [11] (2) p1 q1 ∗ p2 q2 = p1q1 + p2q2 q 2 1 + q 2 2 , or, in the reduced form, (3) p1 q1 ∗ p2 q2 = p q , p = p1q1 + p2q2 p2q1 − p1q2 , q = q 2 1 + q 2 2 p2q1 − p1q2 . p2 q2 p1 q1 p ′ q ′ = p1q1+p2q2 q 2 1+q 2 2 p ′ = p1q1+p2q2 p2q1−p1… view at source ↗
Figure 2
Figure 2. Figure 2: Conway topograph TM of Markov fractions. Juxtaposition of these two topographs establishes the bijection [11] (4) µ : Q ∩ [0, 1] → MF R, where MF R is the set of Markov fractions between 0 and 1/2, which is a fundamental domain of the natural action on the set MF of all Markov fractions by the integer affine group Aff1(Z) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Part of the Conway topograph with Markov fractions. Proposition 2.1. The numbers on a part of the Markov fraction tree shown above satisfy the following relations: (6) p2q3 − p3q2 = q1, p3q1 − p1q3 = q2, (7) p2q1 − p1q2 = q 2 1 + q 2 2 q3 = 3q1q2 − q3, (8) p ′ 1 = p2q2 + p3q3 q1 , q′ 1 = q 2 2 + q 2 3 q1 , (9) p ′ 2 = p1q1 + p3q3 q2 , q′ 2 = q 2 1 + q 2 3 q2 . In particular, comparing this with the Vieta r… view at source ↗
Figure 4
Figure 4. Figure 4: Markov and Cohn topographs related by the trace map Aigner [1] classified all possible generalisation of the initial matrices A and B, preserving the relation with Markov numbers: (11) A(a) :=  a 1 3a − a 2 − 1 3 − a  , B(a) :=  2a + 1 2 −2a 2 + 4a + 2 5 − 2a  , where a ∈ Z is arbitrary. The original Cohn’s choice corresponds to a = 1. Let (12) Ct(a) =  at mt ct 3mt − at  be the Cohn matrix correspon… view at source ↗
Figure 5
Figure 5. Figure 5: Cohn matrices for a = 0. Proof. We prove the theorem by induction. The statement is clearly true for the Cohn matrices shown on [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Conway topograph TC(2) and its mirror image We can use it to describe the continued fraction expansion of the Markov fractions in the following way.1 Note that there are two ways to write the continued fraction expansion of the rational numbers; we will choose the one with even numbers of partial quotients (e.g. [1, 1] rather than [2]). Consider the mirror Conway topograph T ∗ M of the representatives of M… view at source ↗
Figure 7
Figure 7. Figure 7: Conway topographs T ∗ M and T ∗ con Theorem 4.1. The continued fraction expansions of the Markov fractions in [2, 5/2] is given by the juxtaposition of the Conway topographs T ∗ M and T ∗ con. Proof. From Theorem 4.13 of Aigner [1] the Cohn matrix Ct(a) has the form Ct(a) =  amt + ut mt ct (3 − a)mt − ut  1Boris Springborn informed me that the continued fraction expansion of the Markov fractions can be e… view at source ↗
Figure 8
Figure 8. Figure 8: Markov irrationalities and their continued fraction expansions They are the limits of the corresponding left companions γ − m  p q  of the Markov fraction p q ∈ [2, 5/2], defined by Springborn [11]: lim m→∞ γ − m  p q  = γ  p q  [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
read the original abstract

We show that the Markov fractions introduced recently by Springborn coincide with the index of the Cohn matrices defined by Aigner. This provides a simple concatenation rule for the corresponding continued fractions on the Conway topograph.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the Markov fractions introduced by Springborn coincide with the indices of the Cohn matrices defined by Aigner. It derives from this identification a simple concatenation rule for the associated continued fractions on the Conway topograph, relying on standard continued-fraction identities and topograph concatenation.

Significance. If the identification holds, the result unifies two independent constructions in the Markov spectrum, allowing results and techniques to transfer between the Springborn and Aigner frameworks. The explicit concatenation rule on the Conway topograph is a concrete, usable contribution. The proof's reliance on standard tools (continued fractions, topograph concatenation) without new ad-hoc assumptions or free parameters is a strength.

minor comments (3)
  1. [Abstract] The abstract states the main result clearly but does not indicate the key technical tools (continued-fraction identities and Conway-topograph concatenation) used in the proof; adding one sentence would improve accessibility.
  2. [Introduction] When recalling the definitions of Markov fractions (Springborn) and Cohn-matrix indices (Aigner), include explicit cross-references to the precise statements in the cited works so that readers can verify the objects being identified without consulting the originals.
  3. Notation for the index of a Cohn matrix and for the Markov fraction should be introduced once and used consistently; any temporary symbols used only in intermediate steps should be clearly scoped.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its unifying contribution between the Springborn and Aigner frameworks, and the recommendation for minor revision. The referee accurately summarizes the main result: the coincidence of Markov fractions with Cohn matrix indices, together with the derived concatenation rule on the Conway topograph obtained from standard continued-fraction identities.

Circularity Check

0 steps flagged

No significant circularity; independent identification of prior definitions

full rationale

The paper's central result equates Markov fractions (Springborn) with the index of Cohn matrices (Aigner) via standard continued-fraction identities and Conway topograph concatenation. Both objects originate in independent prior work by different authors; the proof takes those definitions as given and applies only external, non-self-referential tools. No fitted parameters, self-definitional loops, load-bearing self-citations, or renamings of known results appear. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described. The work relies on standard number-theoretic definitions from the cited papers by Springborn and Aigner.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    AignerMarkov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings

    M. AignerMarkov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings. Springer, 2013

    work page 2013

  2. [2]

    C ¸ anak¸ cı, R

    ˙I. C ¸ anak¸ cı, R. SchifflerSnake graphs and continued fractions.Eur. J. Comb.86(2020), article 103081

    work page 2020

  3. [3]

    CohnApproach to Markoff’s minimal forms through modular functions

    H. CohnApproach to Markoff’s minimal forms through modular functions. Ann. Math.61(1955),1-12

    work page 1955

  4. [4]

    ConwayThe Sensual (Quadratic) Form, Carus Mathematical Monographs, Vol.26

    J.H. ConwayThe Sensual (Quadratic) Form, Carus Mathematical Monographs, Vol.26. MAA, 1997

    work page 1997

  5. [5]

    GburOn the minimum of zero indefinite binary quadratic forms.Mathematika25(1)(1978), 94-106

    M.E. GburOn the minimum of zero indefinite binary quadratic forms.Mathematika25(1)(1978), 94-106

    work page 1978

  6. [6]

    GorshkovGeometry of Lobachevskii in connection with certain questions of arithmetic.PhD Thesis, 1953 (in Russian)

    D.S. GorshkovGeometry of Lobachevskii in connection with certain questions of arithmetic.PhD Thesis, 1953 (in Russian). Zap. Nauch. Sem. LOMI67(1977), 39-85. English transl. in J. Soviet Math.16(1981), 788-820

    work page 1953

  7. [7]

    Hacking, Y

    P. Hacking, Y. ProkhorovSmoothable del Pezzo surfaces with quotient singularities.Compositio Math. 146(2010), 169-192

    work page 2010

  8. [8]

    MarkovSur les formes quadratiques binaires ind´ efinies

    A.A. MarkovSur les formes quadratiques binaires ind´ efinies. Mathematische Annalen,15(1879), 381-406;17(1880), 379-399

    work page

  9. [9]

    A. N. RudakovThe Markov numbers and exceptional bundles onP 2.Mathematics of the USSR- Izvestiya32:1(1989), 99–112

    work page 1989

  10. [10]

    Spalding and A.P

    K. Spalding and A.P. VeselovLyapunov spectrum of Markov and Euclid trees.Nonlinearity30(2017), 4428-53

    work page 2017

  11. [11]

    SpringbornThe worst approximable rational numbers.J

    B. SpringbornThe worst approximable rational numbers.J. Number Theory263(2024), 153-205

    work page 2024

  12. [12]

    VeselovMarkov fractions and the slopes of the exceptional bundles onP 2.arXiv:2501.06779

    A.P. VeselovMarkov fractions and the slopes of the exceptional bundles onP 2.arXiv:2501.06779. Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK Email address:A.P.Veselov@lboro.ac.uk

    work page arXiv