The pattern of Mills Mess juggling and a link invariant in 3D?

You can find the juggling pattern of the 3 balls called the Mills Mess juggling here.

And a [simplified animation is found here]

Observation and Attempt: - Notice that for each of the three balls, each ball circulates its own "infinity $\infty$"-shape path. The animation makes the paths less clear, but the above video demonstrates much more clearly.

Question:

• I wonder what would be the 3 closed curve patterns on the x-y plane, from the projection of the spacetime trajectory of 3 balls (3 ball dancing in the 3D space with 1D time) into the 2D plane such as the x-y plane shown on your computer screen?

• Can one identify the curves as the time-trajectory paths of 3 balls (allowing only a slightly deformation away from the x-y plane to the z-axis outside/into your screen direction) in 3D $\mathbb{R}^3$ space as some kind of a link invariant in 3D?

Apparently, the simplest candidates will be (1) a set of Hopf links or (2) the Borromean rings.

• BUT, is it really possible to allow such a link pattern from the 3 curves as the 3 time-trajectory paths of the above juggling 3 balls?

• These three balls can have fixed individual patterns (head makes a small $$\cap$$, body makes an $$\infty$$ and tail makes a big $$\cup$$