Introduction to the Alexander module

1. Preface to abstract (by S. Melikhov): This is expected to be the first in a series of talks devoted to the multi-variable Alexander polynomial and its relations with Kojima's $\eta$-function. In the present talk, Monica will review the very basics of the Alexander module following Rolfsen's textbook Knots and Links and her solution of Rolfsen's exercises. Namely, in the case of knots we will see how to get a presentation of the Alexander module from the Seifert matrix (which will be defined and discussed) and then we will also look at some specific two-component links that happen to admit an easy computation of the Alexander module.

2. Abstract (by M. Cabria): Given a knot $K$ and its Seifert surface $M$, we choose a bicollar $\mathring{M}\times(-1,1)$, by studying $H_1(\mathring{M})$, we define the Seifert form and establish a relation between the former and the intersection form of the bicollared surface. After understanding this relation, we show how to obtain the Alexander matrix from the Seifert form.
To conclude, we also study two-component links and how to find their Alexander invariants.