Unistructurality of cluster algebras of type A˜

It is conjectured by Assem, Schiffler and Shramchenko in [3] 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 Dynkin type or rank 2, see [3]. The aim of this paper is to prove it for algebras of type A˜. We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type A˜. We also prove the automorphism conjecture from [3] for algebras of type A˜ as a direct consequence.