Snake graphs and continued fractions

Ralf Schiffler (UConn)

Abstract :

Snake graphs arise naturally in the theory of cluster algebras (of surface type). Each generator of the cluster algebra is a Laurent polynomial that can be computed by a formula whose terms are parametrized by the perfect matchings of a snake graph.

Forgetting the cluster algebras, one can also study snake graphs as abstract combinatorial objects. It turns out that the set of all snake graphs is in bijection with the set of all continued fractions, such that the numerator of the continued fraction is equal to the number of perfect matchings of the snake graph.

This gives a combinatorial interpretation of continued fractions and a new perspective on cluster algebras. This is joint work with Ilke Canakci.