F-polynomial formula from continued fractions

Abstract

For cluster algebras from surfaces, there is a known formula for cluster variables and F-polynomials in terms of the perfect matchings of snake graphs. If the cluster algebra has trivial coefficients, there is also a known formula for cluster variables in terms of continued fractions. In this paper, we extend this result to cluster algebras with principal coefficients by producing a formula for the F-polynomials in terms of continued fractions.