F-polynomial formula from continued fractions

TL;DR

This paper extends the formula for F-polynomials in cluster algebras with principal coefficients to include continued fractions, building on known results for trivial coefficients and perfect matchings.

Contribution

It introduces a new formula for F-polynomials in cluster algebras with principal coefficients using continued fractions, generalizing previous formulas.

Findings

Derived a continued fraction formula for F-polynomials with principal coefficients

Unified the understanding of cluster variables across different coefficient types

Extended known combinatorial formulas to a broader class of cluster algebras

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.