Cartesian parametrization:  p, q, r coprime positive integers.

The Lissajous knots are the knots associated to the 3D Lissajous curves when they are closed and without double point.

As is proved in the above article by Jones and Przytycki, they also are the knots associated to the trajectories of a ball (not subject to gravity) in a parallelepipedic billiard, or even a cubic one (imagine a glass box).

It can be proved that for all values of p, q, r, there exist values of j and y such that the knot is trivial, and that certain knots such as the trefoil knot are not Lissajous knots.

When p and q are coprime,  and , we get the rectangular billiard knot of type (p,q), with r crossings.

For example, the 3 Lissajous knots represented above are parametrized by: ,  and .
The first one is the 4th prime knot with 7 crossings, written 7₄, and more generally, the curve  is, when n is not a multiple of 3, a knot with 5n – 3 crossings with above/below alternate passages.
It is similar for the curve  for odd values of n, which is a knot with 3n – 2 crossings.

On this page can be found a list of remarkable Lissajous knots; the canonical parametrization used in this list is .
 

Here is a beautiful case of an open knot:  By connecting the ends, we get a trefoil knot.

We can also generalize to a 3D billiard with a convex polyhedral shape. This way, we get all the possible knots, even if we only consider the billiards with a right regular prism shape (Koseleff and Pecker).

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© Robert FERRÉOL  2018