The Fundamental Group

Abstract

It is in its interaction with the theory of the fundamental group that the theory of knots and links becomes almost a part of the general theory of 3-manifolds. It is the exterior of a link (that is, the closure of the complement in S ³ of a small regular neighbourhood of the link) that is studied, by means of its group, as a compact 3-manifold with torus boundary components. In the theory of 3-manifolds this is a very important example, but perhaps not much more than that. Here the view has been taken that to a mathematician it is the proving of results that brings satisfaction, and that this is particularly important in knot theory, wherein a cheerful punter might be satisfied by a good diagram. However, 3-manifold theory is well documented at length elsewhere ([43], [49]), and other more established treatises on knots have dwelt comprehensively on the relationship between links and the fundamental group. Thus what follows in this chapter is but an essay on this topic. It tries to interpret the Alexander polynomial in terms of the fundamental group and to explain what is available in more detail elsewhere.