Joint work with Pierre-Guy Plamondon.
It is conjectured in Cluster Automorphisms and Compatibility of Cluster Variables by Ibrahim Assem, Ralf Schiffler and Vasilisa Shramchenko that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of rank 2, type Dynkin or type $\tilde{\mathbb{A}}$. The aim of this talk is to prove it for algebras arising from surfaces without punctures. We use the bracelets basis, introduced by Gregg Musiker, Ralf Schiffler and Lauren Williams and the algebraic independence of clusters to prove the result.