Download PDF - Lissajous knots and billiard knotsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Lissajous knots and billiard knots

Authors 1, 2

Affiliations

  1. Mathematics Department, University of California at Berkeley, Berkeley, California 94720, U.S.A.
  2. Department of Mathematics, The George Washington University, 2201 G St., Room 428 Funger Hall, Washington, D.C. 20052, U.S.A.

Abstract

We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2.

Bibliography

  1. [BHJS94] M. G. V. Bogle, J. E. Hearst, V. F. R. Jones and L. Stoilov, Lissajous knots, Journal of Knot Theory and Its Ramifications 3 (1994), 121-140.
  2. [BGKT94] M. Boshernitzan, G. Galperin, T. Kröger and S. Troubetzkoy, Some remarks on periodic billiard orbits in rational polygons, preprint, SUNY Stony Brook, 1994.
  3. [Cro95] P. Cromwell, Borromean triangles in Viking art, Math. Int. 17 (1995), 3-4.
  4. [GKT94] G. Galperin, T. Kröger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys. (1994).
  5. [GSV92] G. Galperin, A. Stepin and Ya. B. Vorobets, Periodic billiard trajectories in polygons: generation mechanisms, Russian Math. Surveys 47 (3) (1992), 5-80.
  6. [HK79] R. Hartley and A. Kawauchi, Polynomials of amphicheiral knots, Math. Ann. 243 (1979).
  7. [KMS86] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. 124 (1986), 293-311.
  8. [Mu71] K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174.
  9. [Pr95] J. H. Przytycki, Symmetry of knots: Lissajous and billiard knots, notes from a talk given at the Subfactor Seminar, U.C. Berkeley, May 1995.
  10. [Re91] A. W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), 171-184.
  11. [Ro76] D. Rolfsen, Knots and links, Publish or Perish, 1976.
  12. [Ta95] S. Tabachnikov, Billiards. The Survey, preprint, 1995.
  13. [Wi53] R. Wichterman, The biology of Paramecium, The Blakiston Company, Inc., New York, Toronto, 1953.
Banach Center Publications
1998/42/1
Export to PBN, Opens in new tab
Pages:
145-163
Main language of publication
English
Published
1998
Exact and natural sciences