The Free Calculus and the Elementary Ideals

Abstract

In the last chapter a method was developed for calculating a presentation of the group of any knot in regular position. Unfortunately, it does not follow, as was pointed out, that it is now a simple matter to distinguish knot groups, and thus knot types. There is no general algorithm for deciding whether or not two presentations define isomorphic groups, and even in particular examples the problem can be difficult. We are therefore concerned with deriving some powerful, yet effectively calculable, invariants of group presentation type. Such are the elementary ideals. In this chapter we shall study the necessary algebraic machinery for defining these invariants. Specialization to presentations of knot groups in Chapter VIII then leads naturally to the knot polynomials. With these invariants we can easily distinguish many knot types.