The growth of torus link groups

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Let G be a finitely generated group with a finite generating set S. For g∈ G, let lₛ(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) = ∑ₙ₌₀∞ aₙ tⁿ, where aₙ is the number of elements of G with lₛ(g)=n. If A(t) can be expressed as a rational function of 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.