Strange Long Knots

Here's another thing I've been thinking about (just this afternoon): consider a strange attractor of a dynamical system - the Lorenz attractor is a good one. Here's a Java-enabled picture.

For the purposes of my question, all we need to consider is that the trajectory never intersects itself, and is parameterized by times t.

A knot K can also be treated as a t-parameterized curve in three-space. However, t lies in some finite interval, or you can think of K as repeatedly traced out over all time. There are lots and lots of nice algebraic and combinatorial invariants you can compute with knots, like Vassiliev invariants, the Jones polynomial, the Alexander polynomial, and so on.

Is there anything useful that can be sussed out about strange attractors by trying to extend what we know about knot invariants to trajectories of dynamical systems?


- posted by Scott @ 7:15 PM