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In his book Singular Points of Complex Hypersurfaces, Milnor quotes the Alexander (multivariate) polynomial of the torus link $T_{p,pq}$ \begin{align} \Delta_{T_{p,pq}}(t_1, \dots, t_p) = \frac{((t_1 \cdots t_p)^{q} - 1)^{p-1}}{t_1 \cdots t_p - 1} \quad p \geqslant 2, q \geqslant 1. \end{align} Here, $T_{p,pq}$ is a link consisting of $p$-linked unknots which have pairwise linking number equal to $q$. Observe that there is a variable for each component of said link. I know that if $p$ and $q$ are coprime, then the Alexander polynomial of the torus knot $T_{p,q}$ is simply \begin{align} \Delta_{T_{p,q}}(t) = \frac{(t^{pq} - 1)(t-1)}{(t^{p} - 1)(t^{q} - 1)}. \end{align} Below is a table of torus links ordered by increasing crossing number.

Torus Links by Crossing Number

Question: What is the Alexander (multivariate) polynomial of the torus link $T_{p,q}$, in general?

What I've tried: If $p$ and $q$ are not coprime, then I know that $\Delta_{T_{p,q}}$ must have $\text{gcd}(p,q)$ variables, one for each component. Based on the characteristic polynomial of the corresponding monodromy operator, I guess that there are two integers $m_1$ and $m_2$ such that \begin{align} \Delta_{T_{p,q}}(t_1, \dots, t_\text{gcd}(p,q)) = t^{m_1} (t-1)^{m_2} \frac{((t_1 \cdots t_{\text{gcd}(p,q)})^{pq / \text{lcm}(p,q)} - 1)^{\text{gcd}(p,q)} }{(t_1 \cdots t_p - 1)(t_1 \cdots t_q - 1)} \quad p \geqslant 1, q \geqslant 1. \end{align} I'm not sure how to proceed from here.

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The computation of the Hosokawa polynomial is sketched in Kawauchi's book, page 98 exercise 7.4.4. Hosokawa is the multivariable alexander polynomial where you specialize all the variables to be a common variable, and re-scale by some standard factor. I haven't computed the multi-variable alexander polynomial of the general torus link but I think it should be fairly doable using the standard CW-structure on the complement. The linear algebra can get hairy. –  Ryan Budney May 14 '12 at 22:32
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For the interim I'll use the following for my calculations: \begin{align} \Delta_{T_{p,q}}(t_1, \dots, t_\text{gcd}(p,q)) = \frac{((t_1 \cdots t_{\text{gcd}(p,q)})^{pq / \text{lcm}(p,q)} - 1)^{\text{gcd}(p,q)} }{(t_1 \cdots t_p - 1)(t_1 \cdots t_q - 1)} \quad p \geqslant 1, q \geqslant 1. \end{align} If I derive absurdities, then I know that it isn't correct.

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