Journal of Knot Theory and Its Ramifications · 1992 · 498 citations · 0 references
Geometric Group TheoryKnot TheoryBinary OperationLoop SpaceAlgebraic CombinatoricsTopological CombinatoricsEnumerative GeometryFundamental RackAlgebraic DistillationTopological Invariant
A rack is a set with a binary operation whose right multiplication is an automorphism, and any codimension‑two link has a fundamental rack that contains more information than its fundamental group; racks provide a complete algebraic framework for studying links, knots, and 3‑manifolds, and have been studied under various names. The paper aims to consolidate rack algebra and demonstrate that the fundamental rack is a complete invariant for irreducible framed links in a 3‑manifold and for the 3‑manifold itself. The authors give examples of computable link invariants derived from the fundamental rack and explain the connection between rack theory and braid theory. They show that the fundamental rack indeed serves as a complete invariant for irreducible framed links in a 3‑manifold and for the 3‑manifold itself.
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 first 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.