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Edit1 (15Sep10). I have not been able to yet access the Jones-Przytycki paper that Pierre cites, but knowing the keywords he kindly provided, I did find related work by Christoph Lamm ("There are infinitely many Lissajous knots" Manuscripta Mathematica 93(1): 29-37 (1997)) that provides useful information:

Edit2 (15Sep10). Here is a little more information on open problems ca. 2000, found in a list by Jozef H. Przytycki in the book that resulted from Knots in Hellas '98:

Edit3 (5Jul11). The question has been answered (affirmatively) in a paper by Pierre-Vincent Koseleff and Daniel Pecker recently (28Jun11) posted to the arXiv: "Every knot is a billiard knot":

Edit4 (4Oct11). A new paper was released by Daniel Pecker, "Poncelet's theorem and Billiard knots," arXiv:1110.0415v1. The context is that, earlier, "Lamm and Obermeyer proved that not all knots are billiard knots in a cylinder," and Lamm conjectured an elliptic cylinder would suffice. Here is the abstract:

Edit1 (15Sep10). I have not been able to yet access the Jones-Przytycki paper that Pierre cites, but knowing the keywords he kindly provided, I did find related work by Christoph Lamm ("There are infinitely many Lissajous knots" Manuscripta Mathematica 93(1): 29-37 (1997)) that provides useful information:

Edit2 (15Sep10). Here is a little more information on open problems ca. 2000, found in a list by Jozef H. Przytycki in the book that resulted from Knots in Hellas '98:

Edit3 (5Jul11). The question has been answered (affirmatively) in a paper by Pierre-Vincent Koseleff and Daniel Pecker recently (28Jun11) posted to the arXiv: "Every knot is a billiard knot":

Edit4 (4Oct11). A new paper was released by Daniel Pecker, "Poncelet's theorem and Billiard knots," arXiv:1110.0415v1. The context is that, earlier, "Lamm and Obermeyer proved that not all knots are billiard knots in a cylinder," and Lamm conjectured an elliptic cylinder would suffice. Here is the abstract: