• Corpus ID: 225073012

On the ordering of the Markov numbers

@article{Lee2020OnTO,
  title={On the ordering of the Markov numbers},
  author={Kyungyong Lee and Li Li and Michelle Rabideau and Ralf Schiffler},
  journal={arXiv: Number Theory},
  year={2020}
}
The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points $(q,p)$ in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can… 
2 Citations
N T ] 7 J un 2 02 1 A comment on the paper “ Continued fractions and orderings on the Markov numbers
We discuss the validity of the proof of the fixed numerator conjecture on Markov numbers, which is the main result of the paper mentioned in the title. 2010 Mathematics Subject Classification: 11A55,

References

SHOWING 1-10 OF 17 REFERENCES
Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that
Cluster algebras and continued fractions
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear
Matrix formulae and skein relations for cluster algebras from surfaces
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any
From Christoffel Words to Markoff Numbers
Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It
The Combinatorics of Frieze Patterns and Markoff Numbers
TLDR
A matchings model is a combinatorial interpretation of Fomin and Zelevinsky's cluster algebras of type A that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns.
Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs
Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula whose terms are parametrized by the perfect matchings
Cluster algebras and triangulated surfaces. Part I: Cluster complexes
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra
Bases for cluster algebras from surfaces
Abstract We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix,
...
...