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SL(2,Z)-matrixizations of generalized Markov numbers

Yasuaki Gyoda, Shuhei Maruyama, Yusuke Sato

TL;DR

This work develops a unified combinatorial and geometric framework for the $k$-generalized Markov numbers via two $SL(2,\ mathbb{Z})$-matrixizations: the $k$-generalized Cohn (GC) triples and the $k$-Markov-monodromy (MM) triples. It establishes explicit correspondences between these matrixizations, demonstrates that their mutation dynamics reproduce the tree structure of all $k$-GM triples, and provides representation-theoretic and geometric interpretations (including fixed-point realizations in the parabolic $k=2$ case and connections to 4-punctured sphere representations). The paper also develops an algorithmic pathway to compute classical Markov numbers from $2$-MM data and extends the combinatorial toolkit using Farey labeling and Hirsch–Hajcontinued fractions to relate $k$-GM numbers to continued fractions and Wahl chains; this yields deep links to surface singularities and deformation theory. Overall, the work builds a robust, multi-faceted bridge between Diophantine combinatorics, hyperbolic geometry, cluster-algebra mutability, and singularity theory, with practical algorithms for computing Markov-type numbers from dynamical and combinatorial data.

Abstract

For $k\geq 0$, a $k$-generalized Markov number is an integer which appears in some positive integer solution to the $k$-generalized Markov equation $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. In this paper, we discuss a combinatorial structure of generalized Markov numbers. To investigate this structure in detail, we use two families of matrices: the $k$-generalized Cohn matrices and the $k$-Markov-monodromy matrices, which are elements of $SL(2, \mathbb{Z})$ whose $(1,2)$-entries are $k$-generalized Markov numbers. We show that these two families of matrices recover the tree structure of the positive integer solutions to the generalized Markov equation, and we give geometric interpretations and a combinatorial interpretation of $k$-generalized Markov numbers. As an application, we provide a computation algorithm of classical Markov number from a one-dimensional dynamical viewpoint. Moreover, we clarify a relation between $k$-generalized Markov numbers and toric surface singularities via continued fractions.

SL(2,Z)-matrixizations of generalized Markov numbers

TL;DR

This work develops a unified combinatorial and geometric framework for the -generalized Markov numbers via two -matrixizations: the -generalized Cohn (GC) triples and the -Markov-monodromy (MM) triples. It establishes explicit correspondences between these matrixizations, demonstrates that their mutation dynamics reproduce the tree structure of all -GM triples, and provides representation-theoretic and geometric interpretations (including fixed-point realizations in the parabolic case and connections to 4-punctured sphere representations). The paper also develops an algorithmic pathway to compute classical Markov numbers from -MM data and extends the combinatorial toolkit using Farey labeling and Hirsch–Hajcontinued fractions to relate -GM numbers to continued fractions and Wahl chains; this yields deep links to surface singularities and deformation theory. Overall, the work builds a robust, multi-faceted bridge between Diophantine combinatorics, hyperbolic geometry, cluster-algebra mutability, and singularity theory, with practical algorithms for computing Markov-type numbers from dynamical and combinatorial data.

Abstract

For , a -generalized Markov number is an integer which appears in some positive integer solution to the -generalized Markov equation . In this paper, we discuss a combinatorial structure of generalized Markov numbers. To investigate this structure in detail, we use two families of matrices: the -generalized Cohn matrices and the -Markov-monodromy matrices, which are elements of whose -entries are -generalized Markov numbers. We show that these two families of matrices recover the tree structure of the positive integer solutions to the generalized Markov equation, and we give geometric interpretations and a combinatorial interpretation of -generalized Markov numbers. As an application, we provide a computation algorithm of classical Markov number from a one-dimensional dynamical viewpoint. Moreover, we clarify a relation between -generalized Markov numbers and toric surface singularities via continued fractions.
Paper Structure (34 sections, 126 theorems, 240 equations, 24 figures)

This paper contains 34 sections, 126 theorems, 240 equations, 24 figures.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Two ways of $SL(2,\mathbb Z)$-matrixization of $k$-GM triples
  4. Relations between $k$-GC triples and $k$-MM triples
  5. Realization of Markov triples as fixed points of $2$-MM triples
  6. Calculation algorithm of $k$-GM number from irreducible fraction
  7. $k$-GM numbers and HJ-continued fractions
  8. Generalized Markov equation
  9. Generalized Markov tree and inverse generalized Markov tree
  10. Generalized Cohn tree and inverse generalized Cohn tree
  11. $k$-generalized Cohn tree
  12. Inverse $k$-generalized Cohn tree
  13. Markov-monodromy tree and Markov-monodromy decomposition
  14. Definition and existence of Markov-monodromy triple
  15. Inverse $k$-Markov-monodromy tree
  16. ...and 19 more sections

Key Result

Theorem 1.3

The following statements hold:

Figures (24)

  • Figure 1: Signed tiles
  • Figure 2: Snake graph associated with $[2,4,2,1]$
  • Figure 3: List of perfect matchings of $\mathcal{G}[5]$
  • Figure 4: Pre-snake graph associated with $t=\dfrac{2}{5}$
  • Figure 9: Signs assigned to triangles and edges in $\mathcal{PG}(2/5)$ when $k=2$
  • ...and 19 more figures

Theorems & Definitions (236)

  • Conjecture 1.1: frobenius
  • Conjecture 1.2: gyo-maru*Conjecture 1.8
  • Theorem 1.3: Proposition \ref{['pr:all-cohn-triple']}, Corollary \ref{['cor:inverse-all-cone']}
  • Theorem 1.4: Proposition \ref{['pr:all-Markov-monodromy-triple']}, Corollary \ref{['pr:all-Markov-monodromy-triple2']}
  • Theorem 1.5: Proposition \ref{['pr:Markov-monodromy-cohn-triple']}, Theorem \ref{['thm:BT-CT2']}
  • Theorem 1.6: Corollaries \ref{['cor:bijection-triples']}, \ref{['cor:tree-iso-BT-CTdag']}
  • Theorem 1.7: Corollary \ref{['cor:Markov-monodromy-decom-algorithm']}
  • Theorem 1.8: Corollary \ref{['parabolic-markov']}
  • Theorem 1.9: Theorem \ref{['thm:PT-LLMT']}
  • Conjecture 1.10
  • ...and 226 more