[zmgsabst]

Sibling makes a good suggestion to read the better physics reply, and my response to it.

Anyons don’t only exist in lower dimensions: because of how knot theory works, we found the simplest cases in a highly confined electrons — but there’s higher dimensions analogs for higher dimensional knots.

You no longer get anyons from particles exchanging position, but rather, from when fields tangle in 2-knots.

However, the rings in a magnetic field form a torus, so tangles in the magnetic field can (theoretically) knot fairly easily, eg from energies swirls in an accretion disk.