Okay, now I see what you mean. They were going in the right direction, but they did mess up. Everywhere they talk about moving a particle in a loop around the o

Okay, now I see what you mean. They were going in the right direction, but they did mess up. Everywhere they talk about moving a particle in a loop around the other, they really should have talked about moving two particles to exchange them. An exchange is of course topologically distinct from doing nothing. A loop is two exchanges in a row -- which is, as you say, topologically trivial in 3D (so the loop phase must be 1, and the exchange phase is +1 or -1, allowing for bosons and fermions).