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The Complexity of Entanglements

By Jean-Luc Thiffeault

Complex entanglements occur everywhere in nature: in human hair, in the threads forming hagfish slime, in carbon nanotubes, in DNA, and in countless other settings. These entanglements involve physical strands, fibers, or polymers. The character of the entanglements is important, as it often affects the macroscopic integrity of the materials.

Figure 1. Model of tangled magnetic field lines representing a solar coronal loop. Courtesy of the Dundee MHD group: http://www.maths.dundee.ac.uk/mhd/.

In certain other cases, the strands being entangled are more ethereal. For instance, researchers have for many years viewed magnetic field lines in solar flux tubes in the spirit of entanglements. They compute the mathematical braid formed by the field lines, and attempt to relate the complexity of the entanglement to the occurrence of solar flares [3, 15]; see Figure 1.

Figure 2. Float trajectories starting in the Labrador Sea. Reprinted with permission from J.-L. Thiffeault, Braids of entangled particle trajectories, Chaos, 20 (2010), 017516. Copyright 2010, AIP publishing LLC.
But entanglements also occur in a more abstract sense––in oceanic float trajectory data, for instance (see Figure 2). Oceanographers would like to know features of the underlying ocean currents, such as how much mixing is taking place. But the float data is a coarse tool: It samples the flow only very sparsely. The scientists often perform local analyses of the data, such as computing an effective diffusivity of individual trajectories. A method of analysis that simultaneously uses all the data would be preferable.

In [12] we analyzed oceanic trajectories by turning them into a mathematical braid. Such a braid is a symbolic sequence of generators describing how the different trajectories cross each other’s paths. We can associate to the braid of trajectories a braiding factor [13], which can be described as follows: Imagine the braid as a rigid structure; at the base hook a rubber band around some of the strands. Now force the rubber band along the rigid braid; the rubber band will typically stretch and grow. The amount by which it grows is the braiding factor. Its time-normalized logarithm, called the braiding exponent, is closely related to the geometric complexity of the braid [6]. If the braid is infinitely long or is repeated infinitely often, the exponent gives the topological entropy of the braid. The latter name has its origin in the realization of the braid in terms of pseudo-Anosov mapping classes of the punctured disk [4].

Figure 3. Poincaré section (stroboscopic map) for a vat of viscous fluid stirred by a rod (the rod’s path is shown in the inset). A chaotic sea and several islands are visible, along with regular orbits near the wall. The squares indicate the initial positions of the trajectories, with the detected non-growing loops drawn in. Reprinted from M.R. Allshouse and J.-L. Thiffeault, Detecting coherent structures using braids, Phys. D, 241 (2012), 95-105, with permission from Elsevier.
Unlike previous measures of complexity, such as the degree of the Alexander polynomial [10], the braiding factor enables us to distinguish between trivial braiding, as with strands twisted like a ribbon, and more intricate braiding, such as the classic pigtail braid (in fact, the pigtail braid is in some sense the “optimal” braid [7]).

What makes the rubber band calculation described above feasible is the fairly recent development of tools for rapidly computing the action of braids on loops [5, 9, 12]. These clever coordinates, called “Dynnikov coordinates,” efficiently encode the homotopy class of a simple closed loop by a small set of integers. They exploit the fact that a two-dimensional curve that winds without intersecting itself quickly runs out of places to go.

If a dynamical system has positive topological entropy, many loops will grow exponentially when acted upon by a braid of trajectories. If the state space of a system is partitioned into several invariant regions, however, there will exist special loops that grow very slowly or do not grow at all—corresponding to boundaries of the invariant regions. Together with Michael Allshouse [2], we developed a numerical method for detecting such slowly growing loops. We applied this method to trajectories generated by a Lagrangian chaotic mixing device; see Figure 3. This is a proposed method for identifying Lagrangian coherent structures [8]. A trajectory-based approach would have the advantage of not requiring the full velocity field to identify the coherent structures.

The method is not perfect: It requires rather long trajectories, and the data from oceanic floats is often too limited. In order to test and refine the method, Margaux Filippi, Tom Peacock, and Séverine Atis are designing a laboratory experiment that we hope to analyze using the braid approach. Marko Budišić is testing the limits of the braid approach using our freely available software package, braidlab [14].

Figure 4. Various “crowds.” Clockwise from top left, in order of decreasing braid complexity: a school of fish; people walking in random directions; athletes participating in a marathon; and pilgrims performing the Hajj. © 2013 IEEE. Reprinted with permission from S. Ali, Measuring flow complexity in videos, in 2013 IEEE International Conference on Computer Vision (ICCV), 1097-1104.

The braid approach does not actually require an underlying flow. The trajectories generated can arise from processes of any type, not necessarily smooth. For instance, we used braids to analyze the dynamics of the grains in a two-dimensional granular medium [11]. An intriguing new application involves crowd dynamics [1]: If we track the position of individuals in a crowd, can we gauge the “complexity” of the crowd motion? S. Ali [1] has computed the braid complexity for several types of crowds, as shown in Figure 4. The Hajj appears at the bottom left (lowest complexity, because the motion of the pilgrims is so well ordered). How useful it is to classify motion this way remains to be investigated, but it is a promising new direction.

This article is based on “Random Braids,” the author's American Mathematical Society Invited Presentation at the 2014 SIAM Annual Meeting. SIAM Presents has made a taped version of the talk available at http://bit.ly/1ypil9d.

Acknowledgments: This research was supported by NSF grant CMMI-1233935.

References
[1] S. Ali, Measuring flow complexity in videos, in 2013 IEEE International Conference on Computer Vision (ICCV), 1097–1104.
[2] M.R. Allshouse and J.-L. Thiffeault, Detecting coherent structures using braids, Phys. D, 241 (2012), 95–105.
[3] M.A. Berger, Coronal heating by dissipation of magnetic-structure, Space Sci. Rev., 28 (1994), 3–14.
[4] P.L. Boyland, H. Aref, and M.A. Stremler, Topological fluid mechanics of stirring, J. Fluid Mech., 403 (2000), 277–304.
[5] I.A. Dynnikov, On a Yang–Baxter map and the Dehornoy ordering, Russian Math. Surveys, 57 (2002), 592–594.
[6] I.A. Dynnikov and B. Wiest, On the complexity of braids, J. Eur. Math. Soc., 9 (2007), 801–840.
[7] M.D. Finn and J.-L. Thiffeault, Topological optimization of rod-stirring devices, SIAM Rev., 53 (2011), 723–743.
[8] G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Phys. D, 147 (2000),  352–370.
[9] J.-O. Moussafir, On computing the entropy of braids, Func. Anal. Other Math., 1 (2006), 37–46.
[10] S.K. Nechaev, Statistics of Knots and Entangled Random Walks, World Scientific, London, 1996.
[11] J.G. Puckett, F. Lechenault, K.E. Daniels, and J.-L. Thiffeault, Trajectory entanglement in dense granular materials, J. Stat. Mech. Theory Exp., 6 (2012), P06008.
[12] J.-L. Thiffeault, Braids of entangled particle trajectories, Chaos, 20 (2010), 017516.
[13] J.-L. Thiffeault, Measuring topological chaos, Phys. Rev. Lett., 94 (2005), 084502.
[14] J.-L. Thiffeault, M.R. Allshouse, and M. Budišić, braidlab: A Matlab package for analyzing data using braids, 2014; http://www.bitbucket.org/jeanluc/braidlab.
[15] A.L. Wilmot-Smith, D.I. Pontin, and G. Hornig, Dynamics of braided coronal loops. I. Onset of magnetic reconnection, Astron. Astrophys., 516 (2010), A5.

Jean-Luc Thiffeault (jeanluc@math.wisc.edu) is a professor of mathematics at the University of Wisconsin-Madison. 

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