i want to compute the Alexanderpolynomial of the torus knot $T_{p,q}$ with $p$ and $q$ coprime. I should work with the groups presentation $G(T_{p,q})=<x,y:x^p=y^q>$ of $T_{p,q}$. I have to use Fox calculus and minors. I have to conclude that the following formula holds: $$\Delta(T_{p,q})=\frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}$$
I have working out the the following things: $X=\{x,y\}$, and we have to work with the free group $F(X)$ and the relation $r=x^py^{-q}$. Then with the homomorphism $\gamma:F(X)\rightarrow G(T_{p,q})$ (thus we make the quotient of the free group with the normal subgroup generated by the relation $r$) and the rule $$\gamma(\frac{\partial ab^{-1}}{\partial x})=\gamma(\frac{\partial a}{\partial x}-\frac{\partial b}{\partial x})$$ we get the following: $$\gamma(\frac{\partial r}{\partial x})=\gamma(\frac{x^p-1}{x-1})$$ $$\gamma(\frac{\partial r}{\partial y})=\gamma(-\frac{y^p-1}{y-1})$$ Question: How can i go further? The homomorphism gives no more information, true?! And also i have to use the Abelization $\mathcal{A}$ of the group and use this as follow to get the matrix $A$: $$\mathcal{A}(\gamma(\frac{\partial r}{\partial x}))$$ $$\mathcal{A}(\gamma(\frac{\partial r}{\partial y}))$$ I know that the Abelization of the group is infinite cyclic, but how to use this to get the final result?
Thank you for help :)