Poncelet's theorem and Billiard knots

TL;DR

This paper demonstrates that any knot can be represented as a billiard ball trajectory inside an elliptic right cylinder, confirming a conjecture and providing a new proof for another, using elliptic functions and Poncelet's theorem.

Contribution

It proves that all knot types can be realized as billiard trajectories in elliptic cylinders, confirming and providing new insights into existing conjectures.

Findings

All knot types can be realized as billiard trajectories in elliptic cylinders.

Provides a new proof of a conjecture by Jones and Przytycki.

Uses elliptic functions and Poncelet's theorem in the proof.

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.