On a family of Caldero–Chapoton algebras that have the Laurent phenomenon

We realize a family of generalized cluster algebras as Caldero–Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero–Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc j on the polygon with orbifold point a representation M(j) of the referred quiver with relations, and by proving that for every triangulation τ and every arc j∈τ the product of the Caldero–Chapoton functions of M(j) and M(j′), where j′ is the arc that replaces j when we flip j in τ equals the corresponding exchange polynomial of Chekhov–Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of E-rigid indecomposable decorated representations of Λ.