Creating Manifolds from Knots

Abstract

So far in this book we have concerned ourselves with the problem of classifying knots (and, of course, links). Intrinsically, this is a knot theoretical problem. This book, however, is twofold in nature and we wish to balance the purely theoretical with some practical applications of knot theory. The various applications of knot theory are discussed in detail in the latter chapters of this book; we would, however, in this chapter like to consider what might be called the classic application of knot theory. One of the most important, even fundamental, problems in algebraic topology is the general classification of manifolds (see Definition 8.0.1 below). In this chapter we will show that it is possible to create from an arbitrary knot (or link) a 3-dimensional manifold (usually shortened to 3-manifold). Hence by studying the properties of knots we can gain insight into the properties of 3-manifolds.