Application of braid statistics to particle dynamics

How, in a simple and forceful way, do we characterize the dynamics of syste ms with several moving components? The methods based on the theory of braid s may provide the answer. Knot and braid theory is a subfield of mathematic s known as topology. It involves classifying different ways of tracing curv es in space. Knot theory originated more than a century ago and is today a very active area of mathematics. Recently, we have been able to use notions from braid theory to map the complicated trajectories of tiny magnetic bea ds confined between two plates and subjected to complex magnetic fields. Th e essentially two-dimensional motion of a head can be represented as a curv e in a three-dimensional space-time diagram, and so several beads in motion produce a set of braided curves. The topological description of these brai ds thus provides a simple and concise language for describing the dynamics of the system, as if the beads perform a complicated dance as they move abo ut one another, and the braid encodes the choreography of this dance. (C) 1 999 Elsevier Science B.V. All rights reserved.