The Multivariable Alexander Polynomial for a Closed Braid

In this paper I give a similar method for finding the multivariable Alexander polynomial of a link L presented as the closure of a braid β. The main ingredient is a readily constructed multivariable version of the reduced Burau matrices. Other versions of ‘coloured’ Burau matrices have been developed, for example by Penne, [4], which can be interpreted as determining linear presentations of suitably extended versions of the braid group. The most useful feature of the matrices which are used here is that they are extremely simple to remember and they give an immediate and very straightforward construction of the Alexander polynomial of a closed braid and axis, leading at once to the polynomial of the closed braid. For a pure braid the resulting matrix is conjugate to a reduced version of the Gassner matrix; the construction given here has the advantage that it applies to any braid which presents the link, and does not require the braid to be rewritten in any special form. An implementation of this calculation by a Maple procedure, which returns the multivariable Alexander polynomial of β̂ given the braid β, was made in early