Cluster algebras and continued fractions

TL;DR

This paper links continued fractions with cluster algebras through snake graphs, providing combinatorial interpretations, new identities, explicit formulas, and asymptotic analysis of cluster algebra elements.

Contribution

It introduces a novel combinatorial realization of continued fractions via snake graphs in cluster algebras, connecting these areas and deriving new identities and formulas.

Findings

Continued fractions are represented as ratios of perfect matchings of snake graphs.

New identities for continuants of continued fractions are derived from snake graph calculus.

Explicit formulas for quotients of cluster algebra elements as continued fractions are established.

Abstract

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 naturally in the theory of cluster algebras. To a continued fraction $[a_1,a_2,\ldots,a_n]$, we associate a snake graph $\mathcal{G}[a_1,a_2,\ldots,a_n]$ such that the continued fraction is the quotient of the number of perfect matchings of $\mathcal{G}[a_1,a_2,\ldots,a_n]$ and $\mathcal{G}[a_2,\ldots,a_n]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First, we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then, we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.