Finding the Alexander polynomial of the following braid closure

Question

How do you find the Alexander polynomial of the closure of the following braid,
$(\sigma_1^{-2}\sigma_2^{-1}\sigma_3^{-1}\sigma_4^{-1}\sigma_5^{-1}...\sigma_{A-1}^{-1})^B$ where $A$ and $B$ are positive integers?
I have found the general form of the Seifert matrix of link, and then tried to use the formula
$$\Delta(t) = \det (V-tV^T) $$ where $V$ is the Seifert matrix and $\Delta(t)$ is the Alexander polynomial.

I therefore tried using MATLAB to calculate the Alexander polynomials of the knots for $A$ and $B$ running from $1$ to $10$. From the results I guessed the general form of the Alexander polynomial, but I do not end up in a rigorous proof.

I am not quite familiar with the computation of Alexander polynomials using other techniques such as the knot group and Jacobian, so I am not sure if those routes would give a good way in attacking the problem.

PS. I did calculate the Jones polynomial quite quickly, but I don't see any help from that.

Answer 1

The braid you mention is an element of $B_A$ I assume? It doesn't really matter if you add strings after this, as taking the braid closure will just add a new unlink for every new string.

In any case, it seems like you would be interested in the Burau representation of the braid group. The Alexander polynomial of a braid closure $\hat{\beta}$ is given by the determinant of the matrix $I-\beta_*$ where $\beta_*$ is the reduced Burau representation of the braid $\beta$. There is a large amount of literature on the Burau representation of a braid going back to the 60s and 70s. I would strongly suggest picking up Birman's book¹ for a good introduction to theory behind the representations of the braid groups and how to calculate such maps.

¹J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.