2016/08/31 by ̛İlke Çanakçı, Ilke Canakci, Ralf Schiffler · 47 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bijection #Cluster (spacecraft) #Cluster algebra #Combinatorics #Connection (principal bundle) #Geometry #Graph #Mathematics #Pure mathematics #Quantum #Quotient #Realization (probability) #math.AC #math.CO #math.NT
paper · pdf · open access · doi:10.1112/s0010437x17007631
published in Compositio Mathematica 154(3), 565-593 (Cambridge University Press) · 28 pages, Extended introduction and bibliography
openalex created_date 2016/09/16 · arxiv created 2016/09/26 · openalex publication_date 2017/12/22 · arxiv updated 2019/02/20 · openalex updated_date 2026/08/05
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 [a1,a2,… ,an] we associate a snake graph G[a1,a2,… ,an] such that the continued fraction is the quotient of the number of perfect matchings of G[a1,a2,… ,an] and G[a2,… ,an] . 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.