JSJ-decompositions of knot and link complements in the 3-sphere

Abstract

This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in the 3-sphere. Formulated in the language of graphs, the result is the construction of a bijective correspondence between the isotopy classes of links in $S^3$ and a class of vertex-labelled, finite acyclic graphs, called companionship graphs. This construction can be thought of as a uniqueness theorem for Schubert's `satellite operations.' We identify precisely which graphs are companionship graphs of knots and links respectively. We also describe how a large family of operations on knots and links affects companionship graphs. This family of operations is called `splicing' and includes, among others, the operations of: cabling, connect-sum, Whitehead doubling and the deletion of a component.