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It's pretty obvious that the space $L$ of unimodular lattices $\Lambda \subset \mathbb C$ is a complement of trefoil knot: $S^3 \setminus T$. Consider a flow $f_t:= \left( \begin{array}{ccc} e^t & 0 \\ 0 & e^{-t} \end{array} \right)$ on $\mathbb C = \mathbb R^2$. It induces flow $F_t$ on $L$. (Of course, there's whole $U(1)$ of flows $e^{i\phi}f_t$.) Let's take a look at orbits; some natural questions arise.

  • (school arithmetics — but I'm bad in arithmetics) Which lattices (besides obvious $(1, i)$) have closed orbits for some flow? Is there something special in corresponding elliptic curves?
  • (should be easy) Take the image of $(0, s)$ and close it up somehow to a knot $U$ (for example, geodesically for any natural metric — Hausdorff distance, or $S^1$ (orbi)bundle on modular curve structure, pick any). Count linking of $U$ with $T$. What's asymtotics for that number? Is it "uniform" in some sense on nonperiodic orbits?
  • (probably hard) What knots can be obtained this way? Can we have some sort of "dynamics of link groups" for $S^3 \setminus (T \cup U_t)$? Can we recover (theoretically or practically) elliptic curve from this sequence of links given up to isotopy/concordance?

I guess that this construction is too natural to not be already developed, but arXiv search gives not very relevant papers.

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Please look at

http://www.josleys.com/articles/ams_article/Lorenz3.htm

  • section 3 on Modular flows and see if that answers your question.
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