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I recently started reading Étienne Ghys slides on knots and dynamics <http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf> which seem very interesting. I know this approach to knots and dynamics is not entirely new, for instance, this is from the 1980's <http://www.math.columbia.edu/~jb/bw-KPO-I.pdf>. This post gave me the impression that it is a promising topic http://terrytao.wordpress.com/2007/08/03/2006-icm-etienne-ghys-knots-and-dynamics/. Two questions: (the second one is pretty general and I don't expect to have a response if someone finds it not too appropriate for this site, although an answer is always tremendously appreciated)

  1. Are there any recent breakthroughs after the release of these Ghys's slides that I should pay attention to?
  2. Can anyone think of a research opportunity related to this topic for a not too long master's thesis project?

In general I am more interested in the theoretical results rather than its physical applications. In particular those involving geometry and topology. Thanks in advance.

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3 Answers 3

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It's not clear when Ghys made the slides to which you have linked. The only date I could find in those was 1963 (referring to the Lorenz equations), which would make the bound on "recent" rather generous. Here's a quick summary of relatively recent concrete activity in this area that might be interesting to you.

In 1983, various reasonable conjectures were made by Birman and Williams (see BW1 and BW2) after considerable experimentation with the Lorenz equation at various parameter values. The basic line of investigation aimed to find answers to the following question:

What types of knots and links can occur as periodic orbits of stable flows on $S^3$?

For instance, it was conjectured in BW1 that no flow supported all knots as periodic orbits. The conjecture stood uncontested for just over a decade, until Ghrist constructed a flow supporting all links as periodic orbits! His paper G is merely six pages long, picture-filled, and eminently readable.

The entire story is summarized in Bob Williams' article W which contains a wealth of other information including references.


This is really not my area of expertise, but here are two potential research problems. I'm not sure how tractable or elementary this would be, but that's the nature of the beast...

Given a triangulated knot $K$, construct small triangulations $T$ of $S^3$ and piecewise-linear flows $\phi:T \to T$ which contain $K$ as a periodic orbit.

And somewhat harder-seeming,

Given a family $F$ of knots, construct a flow on $S^3$ which supports all knots as periodic orbits except the ones in $F$.


References

BW1: Birman and Williams, Knotted periodic orbits in dynamical systems-I: Lorenz's equations, Topology 22 , 1 (1983), 47 - 82.

BW2: Birman and Williams, Knotted periodic orbits II: Knot holders for fibered knots, Low Dimensional Topology, Contemporary Mathematics 20, A.M.S. (1983), 1-60.

G: Ghrist, Flows on $S^3$ supporting all links as orbits, Electronic Research Announcements of the AMS, 1(2), 91-97 (1995).

W: Williams, The universal templates of Ghrist, BAMS 35, No. 2, 145-156 (1998).

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There are certainly plenty of connections between knot theory and dynamical systems. On the fairly physical end of things, there was a recent workshop you might find interesting:

http://www.kitp.ucsb.edu/activities/dbdetails?acro=knots-m12

There are two major threads in this workshop that were talked about. One was knotted/linked defect fields that appear in liquid crystals under appropriate circumstances. The dynamics of these knotted fields is of current interest. The other major thread was knotted vortices, for example in water. Many of the talks were recorded and the KITP has them available on-line if you would like to explore these themes. This is still a fairly active area of research.

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IF you are willing to extend into "braid theory and dynamics", there is quite a bit of activity in the field of "topological fluid mechanics" in last decade.

Some of this work is directed at determining topological entropy lower bounds on classes of dynamical systems arising from fluid mixing application. The use of Thurston-Nielsen theorem via braid theory has proven useful here.

See papers/monographs of P. Boyland, Thiffeault etc. For example: A recent talk by Boyland: http://www.math.fsu.edu/~hironaka/FSUUF/boyland.pdf

Talk by Thiffeault at SIAM: http://www.math.wisc.edu/~jeanluc/talks/siam2013.pdf

Also, check out this article from 2001 laying out open problems in topological fluid mechanics, several of which have knot and braid theoretic flavor.

Some Remarks on Topological Fluid Mechanics: HK MOffatt 2001

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