Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
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

1 Answer 1

up vote 0 down vote accepted

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.