## 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.