The growth of torus link groups

Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$,where$a_n$isthenumberofelementsof$G$with$l_S(g)=n$\mathrm{.}If$A(t)$canbeexpressedasarationalfunctionof$t$,then$G$ is said to have a rational growth function. We calculate explicitly the rational growth functions of $(p,q)$-torus link groups for any $p,q>1.$ As an application, we show that their growth rates are Perron numbers.