Poncelet's theorem and Billiard knots

Daniel Pecker (1, 2)

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 IMJ - Institut de Mathématiques de Jussieu

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.

Mots clés: Poncelet's theorem; Jacobian elliptic functions; Billiard knots; Lissajous knots; Cylinder knots

Domaines: Géométrie algébrique [math.AG]; Topologie géométrique [math.GT]

Daniel Pecker. Poncelet's theorem and Billiard knots. Geometriae Dedicata, 2012, 161 (1), pp.323-333. ⟨hal-00628619⟩