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Cluster algebras and continued fractions

Çanakçı, İlke; Schiffler, Ralf

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İlke Çanakçı

Ralf Schiffler


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 we associate a snake graph such that the continued fraction is the quotient of the number of perfect matchings of and . 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.


Çanakçı, İ., & Schiffler, R. (2018). Cluster algebras and continued fractions. Compositio Mathematica, 154(03), 565-593.

Journal Article Type Article
Acceptance Date Aug 24, 2017
Online Publication Date Dec 22, 2017
Publication Date Mar 31, 2018
Deposit Date Aug 10, 2018
Publicly Available Date Nov 27, 2019
Journal Compositio Mathematica
Print ISSN 0010-437X
Electronic ISSN 1570-5846
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 154
Issue 03
Pages 565-593
Related Public URLs


Accepted Journal Article (496 Kb)

Copyright Statement
This article has been published in a revised form in Compositio Mathematica This version is published under a Creative Commons CC-BY-NC-ND. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © The Authors 2017

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