Topological Entropy of Surface Braids and Maximally Efficient Mixing

Abstract

The deep connections between braids and dynamics by way of the Nielsen-Thurston classification theorem have led to a wide range of practical applications. Braids have been used to detect coherent structures and mixing regions in oceanic flows, drive the design of industrial mixing machines, contextualize the evolution of taffy pullers, and characterize the chaotic motion of topological defects in active nematics. Mixing plays a central role in each of these examples, and the braids naturally associated with each system come equipped with a useful measure of mixing efficiency, the topological entropy per operation (TEPO). This motivates the following questions. What is the maximum mixing efficiency for braids, and what braids realize this? The answer depends on how we define braids. For the standard Artin presentation, well-known braids with mixing efficiencies related to the golden and silver ratios have been proven to be maximal. However, it is fruitful to consider surface braids, a natural generalization of braids, with presentations constructed from Artin-like braid generators on embedded graphs. In this work, we introduce an efficient and elegant algorithm for finding the topological entropy and TEPO of surface braids on any pairing of orientable surface and planar embeddable graph. Of the myriad possible graphs and surfaces, graphs that can be embedded in $\mathbb{R}^2$ as a lattice are a simple, highly symmetric choice, and the braids that result more naturally model the motion of points on the plane. We extensively search for a maximum mixing efficiency braid on planar lattice graphs and examine a novel candidate braid, which we conjecture to have this maximal property.