On the classification and description of quantum lens spaces as graph algebras

We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underset{‾}{m}))$, introduced by Brzezi'nski-Szyma'nski as graph $C^{\ast }$-algebras. We give a new description of $C(L_q^{2n+1}(r;\underset{‾}{m}))$ as graph $C^{\ast }$-algebras amending an error in the original paper by Brzezi'nski-Szyma'nski. Furthermore, for $n\le 3$, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group $r$. This builds upon the work of Eilers, Restorff, Ruiz and S{\o}rensen.