## 6 Citations

On the ordering of the Markov numbers

- Mathematics
- 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…

$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS

- MathematicsForum of Mathematics, Sigma
- 2020

We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The…

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…

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

SHOWING 1-10 OF 35 REFERENCES

Cluster algebras and Jones polynomials

- MathematicsSelecta Mathematica
- 2019

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…

Cluster algebras and Weil-Petersson forms

- Mathematics
- 2003

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of…

Cluster algebras and triangulated surfaces. Part I: Cluster complexes

- Mathematics
- 2006

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…

Cluster expansion formulas and perfect matchings

- Mathematics
- 2008

We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these…

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…

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

- Mathematics
- 2015

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…

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

- Mathematics
- 2013

Approximation of Irrational Numbers.- Markov's Theorem and the Uniqueness Conjecture.- The Markov Tree.- The Cohn Tree.- The Modular Group SL(2,Z).- The Free Group F2.- Christoffel Words.- Sturmian…