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Orderings of k-Markov Numbers

Esther Banaian

TL;DR

The paper extends Frobenius's unicity viewpoint from ordinary Markov numbers to the $k$-Markov family, proving that Aigner's conjectures hold for all $k\ge0$ by developing a cluster-algebra–inspired combinatorial framework. It builds a bridge between Markov-type Diophantine equations and rational labeling, continued fractions, snake graphs, and fence posets, with skein relations providing the key recurrences. A central construction is the $k$-Markov distance, defined via weighted posets associated to arcs in the integer lattice, and shown to realize the $k$-Markov numbers as continued-fraction numerators. The work generalizes prior $k=1$ results and offers a robust, skein-based method to analyze orderings on rational labels, linking discrete geometry with number-theoretic unicity questions. It also lays groundwork for future explorations of how these orderings interact across different $k$ and with broader families of continued-fraction identities.

Abstract

The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures.

Orderings of k-Markov Numbers

TL;DR

The paper extends Frobenius's unicity viewpoint from ordinary Markov numbers to the -Markov family, proving that Aigner's conjectures hold for all by developing a cluster-algebra–inspired combinatorial framework. It builds a bridge between Markov-type Diophantine equations and rational labeling, continued fractions, snake graphs, and fence posets, with skein relations providing the key recurrences. A central construction is the -Markov distance, defined via weighted posets associated to arcs in the integer lattice, and shown to realize the -Markov numbers as continued-fraction numerators. The work generalizes prior results and offers a robust, skein-based method to analyze orderings on rational labels, linking discrete geometry with number-theoretic unicity questions. It also lays groundwork for future explorations of how these orderings interact across different and with broader families of continued-fraction identities.

Abstract

The -Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to . When , this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that -Markov numbers also satisfy Aigner's conjectures.

Paper Structure

This paper contains 11 sections, 24 theorems, 35 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Tools
  3. Rational Labeling
  4. Continued Fractions
  5. Snake Graphs and Fence Posets
  6. Skein Relations
  7. Two Poset Constructions
  8. Markov Distance
  9. Induced Ordering on Rational Numbers
  10. The Generalized Aigner's Conjectures
  11. More Broadly

Key Result

Theorem 1

If $(x,y,z)$ is a Markov triple, so is $(x,y,z')$ where $z'$ is defined by and similarly for $(x',y,z)$ and $(x,y',z)$. Every Markov triple is the result of applying finitely many such moves to the triple $(1,1,1)$.

Figures (4)

  • Figure 1: On the left, we have a snake graph with 6 tiles. The sign function on the snake graph is shown in the middle, with the signs used to compute the shape colored in red. The red signs (doubling the last $-$) tell us that the snake graph has shape $3,2,2$. The fence poset of shape $3,2,2$ is drawn on the right. Here, we label elements with their chronological labels in boxes.
  • Figure 2: The main process in Algorithm \ref{['algo:New']}
  • Figure 3: An example of an arc $\gamma^\mathrm{str}$. This arc consists of 4 line segments and three angles, $\theta_1,\theta_2$ and $\theta_3$. The first two angles are negative whereas the last is positive. We have perturbed the central intersections of the straight line segments to highlight Convention \ref{['conv:StraightenedArcs']}.
  • Figure 4: One case of a straightened arc $\gamma^\mathrm{str} = \gamma$ with endpoints in $A$ and $B$ which is not homotopic to $\gamma_{AB}^L$ or $\gamma_{AB}^R$. We intersect this arc with another, $\delta$, and resolve the intersection, yielding a shorter arc, $\gamma'$, with endpoints $A$ and $B$. See the proof of Lemma \ref{['lem:FinalStep']}.

Theorems & Definitions (60)

  • Conjecture 1: Aigner aigner2013Markov
  • Theorem 1
  • Theorem 2: Theorem 1.1 GyodaMatsushita
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 50 more