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I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$ that has these properties:

(1) When iterated $n$ times starting from some $p$, connecting the points in order with segments and closing last to first,

$$(p, f(p), f^2(p), \ldots, f^n(p), p)$$

results in a simple (non-self-intersecting) closed polygonal cycle $K$.

(2) When $K$ is viewed as a knot, it is highly tangled, e.g., it has large crossing number, or large unknotting number. The tangledness, however defined, should increase with $n$, the faster the better.

(3) These properties should hold for infinitely many $n$.

Expressed differently, I would like a way to generate an infinite variety of increasingly tangled stick knots via a simple function iteration. My requirements are a bit loose, as I just want to simply generate knotty examples. Likely some weaving is known to accomplish this...?
          Tangled Knot
(This question is intellectually related to an earlier question, "Complexity of random knot with vertices on sphere.")

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2 Answers 2

up vote 8 down vote accepted

If you parametrize a torus in $\mathbb{R}^3$ as $(x(u,v),y(u,v),z(u,v))$, $0\le u,v<1$, you can easily generate the torus knot $(3,q)$ (with crossing number $2q$) for $q$ large enough and not divisible by three by letting $f(u,v) = (u+3/q^2 \mod 1,v+1/q \mod 1)$ and $n=q^2$. So you just have to embed a continuum of these tori with $q$ varying continuously, and you'll have a function $f$ that generates every knot $(3,q)$. You can probably also improve on how $n$ grows with $q$.

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@Yoav: Very simple! Not unlike my figure. Thanks! –  Joseph O'Rourke Apr 29 '12 at 11:41

There is a body of work starting with Birman and Williams, and continued by Ghrist, Holmes, and Sullivan, concerning knottedness of closed orbits of flows. The flows they consider are certain Axiom A flows on $S^3$ with interesting basic sets, for example the Lorentz attractor. Williams' paper "Lorentz knots are prime" shows that all the closed orbits of the Lorentz attractor are prime knots. And, being an Axiom A basic set, closed orbits are dense in the Lorentz attractor. So if $f$ is the time $\epsilon$ map of a flow on $S^3$ having the Lorentz attractor as a basic set, then you'll get lots of examples by choosing $p$ on longer and longer closed orbits, as long as you are careful to connect $f^k(p)$ to $f^{k+1}(p)$ by a flow segment. I am guessing that these knots will be more and more complicated as you choose $p$ to have longer and longer orbit.

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Here's a nice recent paper of Birman and Kofman on the subject that also has a good bibliography of previous work : math.columbia.edu/~jb/bk-lorenz-jtop.pdf –  Fred Apr 29 '12 at 4:20
    
About the same line of develeopment: You should look at the slides for Etienne Ghys's ICM talk about "modular knots": umpa.ens-lyon.fr/~ghys/articles/icm.pdf The slides alone are absolutely stunning, and the talk (which I heard elsewhere) is one of the best ever. You might also enjoy this paper: ams.org/featurecolumn/archive/lorenz.html (Lorenz and modular flow: a visual introduction, Ghys and Leys, Feature Column, AMS, November 2006.) –  Alex Eskin Apr 29 '12 at 7:47
    
@Lee, Fred, Alex: Great references! Indeed I was thinking of attractors, but knew nothing of this literature. Thanks! –  Joseph O'Rourke Apr 29 '12 at 11:42

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