We can use Braid Theory - a sub-discipline of Knot Theory - to talk about juggling. Juggling patterns are usually in a single plane, so imagine juggling while walking backwards with the balls leaving glowing trails. Those trails form a braid.

http://en.wikipedia.org/wiki/Braid_theory

There are "invariants" for braids, which can then be used to distinguish juggling patterns. That can be used to show that the 3 Cascade is in some sense fundamentally different from Mills Mess. Mills Mess is actually the "Unbraid", something that's rather non-obvious.