In case you thought I had given up on knot theory

August 4, 2007 in geekiness, links, math, thought

I guess it might actually be pretty cool. Professional mathematician and fellow wordpress.com blogger Terrance Tao has a really cool post summarizing a talk from the 2006 ICE about the marriage of knot theory and dynamical systems. Specifically, he talks about the use of limits of knots (whatever that means…) as a way of understanding the dynamics of the Lorenz attractor (pictured above), a quite pretty-looking “strange attractor” in \mathbb{R}^3. However, that analysis is based on interpreting the periodic orbits of the system as knots (when taken alone) or links (when taken together), and in general this doesn’t work. Some systems are positively riddled with periodic orbits, and some have practically none at all, so people started trying to interpret nonperiodic orbits as knots/links. This is accomplished by taking a sort of ergodic-flavored limit of the series of knots formed by picking a starting point x in the space and a long time T. The knots then come from tracing out the path of x as it moves around for time T and then connecting the final point \phi^T(x) to x with a line segment to close the loop and form a knot.

The ergodic flavor comes from how the limit is taken and is better explained, I think, using the concrete example that Tao gives in his post. The goal is to understand the helicity of a 3-D dynamical system, which according to Wikipedia is “the extent to which corkscrew-like motion occurs.” The idea is to take two random starting points, x_1 and x_2 and a long time T. Form the first knot K_{1,T} as before, by tracing out the path through which x_1 moves in time T and joining the final point \phi^T(x_1) to the starting point x_1 to form a closed loop. Do the same thing with x_2 to get another knot, K_{2,T}.

Now we have two knots, K_{1,T} and K_{2,T} that are probably looped together in interesting ways. We’ll call link consisting of the two of them taken together L_{T} (in math speak, L_{T} = K_{1,T} \cup K_{2,T}). In knot theory, the linking number of a link is a measure of how tangled up the separate loops of the link are, and is defined so that it’s the same no matter how you look at the link.

Taking a step back to look at the big picture, we can see that if a body of water is corkscrewing as it moves, the paths traced out by two particles in the water (K_{1,T} and K_{2,T} here) will be wound together in a DNA-ish double spiral. Furthermore, the faster the water is corkscrewing, the more tightly coiled this spiral will be. Since the degree of spiralling is measured here by the linking number of L_{T}, by dividing that linking number by the time T we can get a measure of the average rate of corkscrewing over this time interval. Taking the limit of this normalized linking number as T\rightarrow\infty and “averaging” over all choices of starting points x_1 and x_2 gives us a measure of how much corkscrewing there is in the motion defined by \phi.

Taking this sort of limit (of some quantity that cumulatively changes over time, averaged for that time and across the entire space) is the bread and butter of ergodic theory (the study of dynamical systems). The difficult thing about studying systems this way is making sure that these limits actually exist, and this is what most of the long proofs from the introductory stuff I studied were aimed at doing. Making sure a limit exists is usually done by hand-crafting the terms of the limit to make sure everything works, so it would be interesting if ideas from a totally different discipline (here, knot theory) could be used, essentially unchanged, as the basis for these finicky limits.

And, indeed, in this case the technique does work: the limit described above, also called the time-average of the linking number, is exactly equal to the helicity of the flow \phi. To me (and, I imagine, to other mathematicians), this is interesting because it suggests that there is enough similarity between the structure of space as formalized by dynamical systems and as formalized by knot theory to make the one expressible in terms of the other. So, for all my frustration at the seeming arbitrariness and lack of rigor of knot theory, it is somehow a close cousin of ergodic theory, the subject of probably my favorite math class I’ve ever taken.

I’m endlessly fascinated by these higher-level structural isomorphisms, whether they’re found in mathematics, human knowledge representation, language, or the natural world. Probably unsurprisingly, I suspect that the awareness and embodiment of these sorts of structure is what makes thinking (or even living) things what they are. This doesn’t mean that I’m in the symbolic-representationalist camp of cognitive science, which holds that abstract structures and rules are represented explicitly in human representations of information. The fact that structural similarities (like the ergodic theory-knot theory one discussed here) often happen seemingly by accident is, I think, evidence against such explicitly representational systems. If you’re a long-time (and rather brave) reader, you might remember my rantings about the importance of emergence in science, and more generally ontology. I’m not sure right now how this relates, but I know that it does, so I’ll just leave you to mull over that until I get my hairbrained mind in order.

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