In recent years, topological concepts have yielded valuable insights into the long standing problem of laminar fluid mixing. Topologically complex stirring protocols are typically far superior to topologically simple protocols, guaranteeing chaotic advection of fluid particles and the associated exponential dilation of material elements. Furthermore, topological approaches to mixer design are typically intuitive and insensitive to precise geometry or fluid properties. However, results to date

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... ve been limited to two dimensional flows (for example, batch stirrers in food or polymer manufacturing) and quasi three dimensional protocols (for example, continuous flow micromixers). Motivated by a simple stretching and folding argument that works well in two dimensions, we propose a topological approach to fully three dimensional fluid mixing. A transition matrix is derived to describe the mapping induced by a three dimensional 'braid' on area elements,