Lissajous knots and billiard knots

Vaughan Jones; Józef Przytycki

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 145-163
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Jones, Vaughan, and Przytycki, Józef. "Lissajous knots and billiard knots." Banach Center Publications 42.1 (1998): 145-163. <http://eudml.org/doc/208802>.

@article{Jones1998,
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.},
author = {Jones, Vaughan, Przytycki, Józef},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {145-163},
title = {Lissajous knots and billiard knots},
url = {http://eudml.org/doc/208802},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Jones, Vaughan
AU - Przytycki, Józef
TI - Lissajous knots and billiard knots
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 145
EP - 163
AB - 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.
LA - eng
UR - http://eudml.org/doc/208802
ER -

References

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  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. Zbl0777.58031
  6. [HK79] R. Hartley and A. Kawauchi, Polynomials of amphicheiral knots, Math. Ann. 243 (1979). Zbl0394.57009
  7. [KMS86] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. 124 (1986), 293-311. Zbl0637.58010
  8. [Mu71] K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174. Zbl0206.25603
  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. Zbl0847.57013
  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. 

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