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It is conjectured by Ibrahim Assem, Ralf Schiffler and Vasilisa Shramchenkoin "Cluster Automorphisms and Compatibility of Cluster Variables" that everycluster algebra is unistructural, that is to say, that the set of clustervariables determines uniquely the cluster algebra structure. In other words,there exists a unique decomposition of the set of cluster variables intoclusters. This conjecture has been proven to hold true for algebras of typeDynkin or rank 2 by Assem, Schiffler and Shramchenko. The aim of this paper isto prove it for algebras of type ${\widetilde{\mathbb{A}}}$. We usetriangulations of annuli and algebraic independence of clusters to proveunistructurality for algebras arising from annuli, which are of type${\widetilde{\mathbb{A}}}$. We also prove the automorphism conjecture fromAssem, Schiffler and Shramchenko for algebras of type${\widetilde{\mathbb{A}}}$ as a direct consequence.

Unistructurality of cluster algebras of type ${\widetilde{\mathbb{A}}}$