A closed contact cycle on the ideal trefoil

TL;DR

This paper investigates contact cycles on the ideal trefoil, revealing a numerically suggested nine-step periodic billiard cycle that acts as an attractor for all contact sequences.

Contribution

It introduces the concept of billiards on the ideal trefoil and provides numerical evidence for a specific closed contact cycle as a stable attractor.

Findings

A nine-step periodic billiard cycle on the ideal trefoil is suggested.

All contact sequences tend to converge to this cycle.

Numerical simulations support the stability of the cycle.

Abstract

Numerical computations suggest that each point on a certain optimized shape called the ideal trefoil is in contact with two other points. We consider sequences of such contact points, such that each point is in contact with its predecessor and call it a billiard. Our numerics suggest that a particular billiard on the ideal trefoil closes to a periodic cycle after nine steps. This cycle also seems to be an attractor: all billiards converge to it.