The Alexander Polynomial

Abstract

The Alexander polynomial of an oriented link is, like the Jones polynomial, a Laurent polynomial associated with the link in an invariant way. The two polynomials give different information about the geometric properties of knots and links. The Alexander polynomial will, for example, give a lower bound for the genus of a knot, but it is not as useful as the Jones polynomial for investigating the required number of crossings in a diagram. The Alexander polynomial will later, in Theorem 8.6, be described combinatorially in terms of diagrams in a way that parallels Proposition 3.7, but the real interest of this invariant is that, in contrast to the Jones polynomial, it has a long history [3] and is well understood in terms of elementary homology theory. The homology approach to the Alexander polynomial, which will now be explained, describes it as a certain invariant of a homology module. To appreciate this, a little information about presentation matrices of modules is needed. There follows, then, a basic discussion of this topic, aimed at obtaining results rapidly. It may be neglected by the cognoscenti.