A Discussion of Higher-Order Alexander Modules

We will discuss a technique to distinguish knots via an algebraic invariant, namely the Alexander module, which is a module associated to the universal abelian cover of the complement of a knot. We will present a generalization of the Alexander module by constructing modules corresponding to different regular covering spaces of the knot complement. We will present a numerical invariant that can be extracted from these modules, and a procedure by which we can compute it in practice in the case of the first higher-order Alexander module. Finally, we will argue that this procedure could feasibly be followed algorithmically, and so a computer could be programmed to compute this numerical invariant.