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Poncelet’s theorem and billiard knots

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Abstract

Let D be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in D. This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi’s proof of Poncelet’s theorem by means of elliptic functions.

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References

  1. Berger M.: Geometry I, II. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  2. Birkhoff G.: On the periodic motions of dynamical systems. Acta Math. 50(1), 359–379 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bos H.J.M., Kers C., Oort F., Raven D.W.: Poncelet’s closure theorem. Expo. Math. 5, 289–364 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Bogle M.G.V., Hearst J.E., Jones V.F.R., Stoilov L.: Lissajous knots. J. Knot Theory Ramif. 3(2), 121–140 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boocher, A., Daigle, J., Hoste, J., Zheng, W.: Sampling Lissajous and Fourier knots. arXiv:0707.4210 (2007)

  6. Cromwell P.: Knots and Links. Cambridge University press, Cambridge (2004)

    Book  MATH  Google Scholar 

  7. Darboux G.: Principes de Géométrie Analytique. Gauthier-Villars, Paris (1917)

    MATH  Google Scholar 

  8. Dehornoy, P.: A billiard containing all links. C. R. Acad. Sci. Paris, Sér. I 349, pp. 575–578 (2011)

  9. Flatto L.: Poncelet’s Theorem, Chapter 15 by Tabachnikov, S. American Mathematical Society, Providence (2009)

    Google Scholar 

  10. Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, Oxford (1960)

    MATH  Google Scholar 

  11. Jacobi C.G.J.: Ueber die Anwendung der elliptischen Trancendenten auf ein bekanntes Problem der Elementargeometrie. Journal für die reine und angewandte Mathematik (Crelle’s Journal) 3, 376–389 (1828)

    Article  MATH  Google Scholar 

  12. Jones V.F.R., Przytycki J.: Lissajous knots and billiard knots. Banach Cent. Publ. 42, 145–163 (1998)

    MathSciNet  Google Scholar 

  13. Koseleff P.-V., Pecker D.: Chebyshev knots. J. Knot Theory Ramif. 20(4), 1–19 (2011)

    Article  MathSciNet  Google Scholar 

  14. Koseleff, P.-V., Pecker, D.: Every knot is a billiard knot, preprint. arXiv:1106.5600vl (2011)

  15. Lamm, C.: Zylinder-Knoten und symmetrische Vereinigungen. Dissertation, Universität Bonn, Mathematisches institut, Bonn (1999)

  16. Lamm C.: There are infinitely many Lissajous knots. Manuscripta Math. 93, 29–37 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lamm, C.: Fourier knots. English translation of a part of Zylinder-Knoten und symmetrische Vereinigungen. Dissertation, Universität Bonn, Mathematisches institut, Bonn (1999) [preprint]

  18. Lamm C., Obermeyer D.: Billiard knots in a cylinder. J. Knot Theory Ramif 8(3), 353–366 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Laurent, H.: Sur les résidus, les fonctions elliptiques, les équations aux dérivées partielles, les équations aux différentielles totales, appendix in the book by F. Frenet, Recueil d’exercices sur le calcul infinitésimal, 5e éd. Gauthier-Villars, Paris (1891)

  20. Levi M., Tabachnikov S.: The Poncelet grid and billiards in ellipses. Am. Math. Mon. 114(10), 895–908 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Manturov V.O.: A combinatorial representation of links by quasitoric braids. Eur. J. Combin. 23(2), 207–212 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. O’Rourke, J.: Tying knots with reflecting lightrays. http://mathoverflow.net/questions/38813/ (2010)

  23. Pecker, D.: Poncelet’s theorem and Billiard knots, preprint. arXiv:1110.0415vl (2011)

  24. Poncelet, J.-V.: Traité des propriétés projectives des figures. Paris (1822)

  25. Przytycki, J.: Symmetric knots and billiard knots, chapter 20. In: Stasiak, A., Katrich, V., Kauffman, L. (eds.) Ideal Knots, Series on Knots and Everything, vol. 19, pp. 374–414. World Sientific, River Edge, NJ (1999)

  26. Samuel P.: Géométrie projective. P.U.F., Paris (1986)

    MATH  Google Scholar 

  27. Steinhaus H.: Mathematical Snapshots, 2nd edn, chapter X. Oxford University Press, London (1950)

    Google Scholar 

  28. Tabachnikov S.: Geometry and Billiards. American Mathematical Society, Providence, RI (2005)

    MATH  Google Scholar 

  29. Whittaker E.T., Watson G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1935)

    Google Scholar 

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  1. Faculté de Mathématiques, U. Pierre et Marie Curie (Paris 6), Paris, France

    Daniel Pecker

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  1. Daniel Pecker
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Correspondence to Daniel Pecker.

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Pecker, D. Poncelet’s theorem and billiard knots. Geom Dedicata 161, 323–333 (2012). https://doi.org/10.1007/s10711-012-9708-2

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  • DOI: https://doi.org/10.1007/s10711-012-9708-2

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