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Frieze patterns (in the sense of Conway and Coxeter [6]) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed continuant polynomial to define a new family of friezes, called c−friezes, which generalises frieze patterns. Having in mind the cluster algebras of finite type, we identify a necessary and sufficient condition for obtaining periodic c−friezes. Taking into account the Laurent phenomenon and the positivity conjecture, we present ways of generating c−friezes of integers and of positive integers. We also show some specific properties of c−friezes.

It is conjectured by Assem, Schiffler and Shramchenko in [ASS14a] 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 type Dynkin or rank 2 [ASS14a]. 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 [ASS14a] for algebras of type A as a direct consequence.

We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant associated to an $\textrm{SL}_3$ web diagram, with respect to a particular term order.