Racks and Links in Codimension 2 IntroductionRACKS AND LINKS IN CODIMENSION TWOROGER

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3{manifolds, and also for the 3{manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this rst paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3{manifold and for the 3{manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids. This is the rst of a series of papers by the authors. More papers, some in collaboration with Brian Sanderson, are in preparation. In these papers we shall study a natural algebraic theory, strongly connected with the theories of groups, group presentations and crossed modules. This is the theory of racks. A rack is a set with a binary operation satisfying two simple laws which are the algebraic distillation of two of the Reidemeister moves (the 2 and 3 moves). Racks have been variously studied by previous authors under a variety of names (including rack) and using a variety of diierent notations and terminology. We shall give a summary of this previous work shortly. One of the aims of this paper is to attempt to establish a uniform set of conventions for notation and terminology in this subject. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3{manifolds, and also for the 3{manifolds themselves. Included in this