JSJ - decompositions of knot and link complements in S 3

This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in S3 . Formulated in the language of graphs, the result is the construction of a bijective correspondence between the isotopy classes of links in S3 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. AMS Classification numbers Primary: 57M25 Secondary: 57M50, 57M15