In recent years, starting with the work of Boyland et al. [2,
3], methods from topology and braid theory have been applied to
the analysis of mixing in flows with great success. Consider a
mixing device consisting of rods moving in a two-dimensional
viscous fluid. If the position of the rods is plotted in a
three-dimensional graph, with time the vertical axis, one
obtains a “spaghetti plot” of the world lines of the rods,
similar to the plot shown here. Because the rods are material
objects, the world lines cannot intersect each other. The
resulting graph thus describes a braid, in the mathematical
sense of a bundle of strands that are not allowed to cross each
other. The braid formed by the world lines of a rod and several
periodic islands in a mixing device. The analysis of this braid
yields important information about the mixing properties of the
flow. For instance, it can be shown that if the braid possesses
a positive topological entropy, then there must exist a region
in the surrounding fluid that exhib