The object of this paper is
to show how to use the Burau representation of the Artin braid group to
calculate some invariants of an oriented link in 𝕊3. More precisely, we obtain
generators and relations for the Alexander module, and
a unimodular (−t)-Hermitian form on the torsion submodule of the
Alexander module (see below for a precise statement).
Scaling our form by (1 −t−1) yields a Hermitian form which, for knots, is probably the
Blanchfield form. If so, it would then follow from Trotter that the S-equivalence class
of the Seifert form of a knot can be computed from the Burau representation. Even if
this form is the Blanchfield form for knots, the situation for links is less clear because
(1 − t−1) need not be invertible in the endomorphism ring of the Alexander
module.
