Braids and Dynamics

This short book considers an unusual combination of algebraic topology, dynamics and applications to the mixing of fluids. Probably not many mathematics texts use a taffy puller in a primary example. This one does, and it incorporates interesting connections to dynamics on surfaces, the algebraic topology of braids and applications to mixing.

 

This book is part of Springer’s “Reviews and Tutorials” series. In relatively few pages it moves fast from topic to topic, quickly introducing and connecting topics in dynamics and algebraic topology. The author suggests that it is aimed at beginning graduate students, advanced undergraduates and practicing mathematicians looking to learn something of the topic.

 

A taffy puller, for those who have never seen one, is a machine for pulling taffy, a chewy candy made by heating a mixture of ingredients with sugar, that must be pulled – stretched and folded – to get air bubbles into the taffy, which gives it a softer and chewier texture.

 

The author begins with a simple motivating example. What happens when black ink is gently mixed into a clear viscous fluid using three stirring rods that follow figure-eight stirring patterns? Results depend on the stirring pattern of the relative motion – clockwise or counterclockwise – of the three rods. The operation of the taffy puller parallels the motion of the rod stirring devices. A good taffy puller creates “Figure 8” motions with its mixing and stretching rods.

 

Braids appear as trajectories of the tips of the rods. The author calls braids “particle dances”, and he follows Artin’s development of the theory of braids. Much of what the author presents is intended to show the connection of braids to dynamics, and particularly to the Thurston-Nielsen classification of diffeomorphisms on surfaces.

 

On the way to a discussion of the deep results of Thurston-Nielsen, the reader encounters a good deal of topology, dynamical systems theory and their relationships. In particular, the concept of topological entropy captures the exponential rate of stretching material curves due to the motion of the points forming the braid. It is an important measure of mixing. It is curious that many of the taffy mixing machines developed in the 19th century have movements consistent with braids of high mixing efficiency.

 

The author also briefly considers the role of braids in data analysis of trajectories of dynamical systems. There is even a description of software for manipulating braids and loops.

 

Several intriguing ideas are presented here very quickly, and it takes some effort to make all the pieces fit together.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.