Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we identify each cluster variable with its snake graph, and interpret relations among the cluster variables in terms of these graphs. In particular, we give a new proof of skein relations of two cluster variables. arXiv:1209.4617v2 [math.RT]
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 of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. We advance the study of abstract snake graphs by introducing the notions of abstract band graphs, self-crossings of abstract snake graphs as well as their resolutions. We show that there is a bijection between the set of perfect matchings of crossing or self-crossing snake graphs and the set of perfect matchings of the resolution of the crossing. In the situation where the snake and band graphs are coming from arcs and loops in a surface without punctures, we obtain a new proof of skein relations in the corresponding cluster algebra.
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.Comment: 28 pages, Extended introduction and bibliograph
We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points on the boundary is one, and we show that the cluster algebra is equal to the upper cluster algebra in this case. arXiv:1407.5060v2 [math.RT]
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