Imagine Math pp 169-174 | Cite as

The Many Faces of Lorenz Knots

  • Marco Abate


One of the greatest pleasures in doing mathematics (and one of the surest signs of being onto something really relevant) is discovering that two apparently completely unrelated objects actually are one and the same thing. This is what Étienne Ghys, of the École Normale Superieure de Lyon, did a few years ago (see [1] for the technical details), showing that the class of Lorenz knots, pertaining to the theory of chaotic dynamical systems and ordinary differential equations, and the class of modular knots, pertaining to the theory of 2-dimensional lattices and to number theory, coincide. In this short note we shall try to explain what Lorenz and modular knots are, and to give a hint of why they are the same. See also [2] for a more detailed but still accessible presentation, containing the beautiful pictures and animations prepared by Jos Leys [3], a digital artist, to illustrate Ghys’ results.


Periodic Orbit Modular Group Left Wing Integer Coefficient Complex Logarithm 
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  1. 1.
    É. Ghys, Knots and dynamics. In: International Congress of Mathematicians. Vol. I: European Mathematical Society, Zürich, pp. 247–277, 2007.Google Scholar
  2. 2. article/Lorenz3.htm
  3. 3.
  4. 4.
    C. Adams, The knot book. American Mathematical Society, Providence RI, 2004.MATHGoogle Scholar
  5. 5.
  6. 6.
    E. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141, 1963.CrossRefGoogle Scholar
  7. 7. Course/Lesson1/Demo8.html
  8. 8.
    J. Birman, R. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations. Topology 22, 47–82, 1983.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    J. Birman, I. Kofman, A new twist on Lorenz links. J. Topology 2, 227–248, 2009.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    J-P. Serre, A course in arithmetic. Springer-Verlag, Heidelberg, 1973.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Marco Abate
    • 1
  1. 1.Department of MathemathicsUniversity of PisaItaly

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