## 9 Citations

On the Markov numbers: Fixed numerator, denominator, and sum conjectures

- MathematicsAdv. Appl. Math.
- 2021

Convexity and Aigner's Conjectures

- Mathematics
- 2021

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic x + y + z − 3xyz = 0. A classical topic in number theory, these numbers…

Boundary slopes for the Markov ordering on relatively prime pairs

- MathematicsAdvances in Mathematics
- 2022

Expansion Posets for Polygon Cluster Algebras

- Mathematics
- 2020

Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some…

Relation Between f-Vectors and d-Vectors in Cluster Algebras of Finite Type or Rank 2

- Mathematics
- 2019

We study the $f$-vectors, which are the maximal degree vectors of $F$-polynomials in cluster algebra theory. When a cluster algebra is of finite type or rank $2$, we find that the positive…

N T ] 7 J un 2 02 1 A comment on the paper “ Continued fractions and orderings on the Markov numbers

- Mathematics
- 2021

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,…

N T ] 2 3 M ay 2 02 1 A comment on the paper “ Continued fractions and orderings on the Markov numbers

- Mathematics
- 2021

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,…

On the monotonicity of the generalized Markov numbers

- Mathematics
- 2022

. Using the Markov distance and Ptolemy inequality introduced by Lee-Li-Rabideau-Schiﬄer [10], we completely determine the monotonicity of the generalized Markov numbers along the lines of a given…

Hernandez-Leclerc modules and snake graphs

- Mathematics
- 2020

In 2010, Hernandez and Leclerc studied connections between representations of quantum affine algebras and cluster algebras. In 2019, Brito and Chari defined a family of modules over quantum affine…

## References

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We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge…

The Combinatorics of Frieze Patterns and Markoff Numbers

- MathematicsIntegers
- 2020

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.

From Christoffel Words to Markoff Numbers

- Mathematics
- 2018

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…

Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings

- Mathematics
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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

- MathematicsCompositio Mathematica
- 2017

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…

Concrete mathematics - a foundation for computer science

- Education
- 1989

From the Publisher:
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid…

Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

- Mathematics
- 2014

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…

On cluster algebras from unpunctured surfaces with one marked point

- Mathematics
- 2014

We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points is one, and we show that the cluster algebra is equal to the…