In the study of noncommutative geometry many classical spaces have been given a quantum analogue. An example is quantum lens spaces, which are defined as fixed point algebras of the quantum sphere by Vaksman and Soibelman under the actions of finite cyclic groups. Hong and Szymański described both quantum spheres and quantum lens spaces, whose weights are all coprime to the order of the acting group, as graph $C^*$-algebras. Later on Brzeziński and Szymański generalised the result to include all quantum lens spaces. Unfortunately, as pointed out by Efren Ruiz, their description is not always correct.

In this talk, I will first present how quantum lens spaces can be described as graph $C^*$-algebras. Then, I will apply the classification result of graph $C^*$-algebras over finite graphs by Eilers, Restorff, Ruiz and Sørensen to obtain a number-theoretic invariant for quantum lens spaces of dimension $7$, where all but a single weight are coprime to the order of the acting group. This is based on joint work with Thomas Gotfredsen.