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This is an exercise in many topology books. Here is a reference with a complete proof: Look up Example 9.15 in the book "Knots" by G. Burde and H. Zieschang. The Jacobian of the presentation $G(T_{p,q})=\langle x,y \mid x^py^{-q}\rangle$ is computed. It is $$ \left( \frac{t^{pq}-1}{t^q-1}, -\frac{t^{pq}-1}{t^p-1}\right). $$ The greatest common divisor of it is the Alexander polynomial.