Braiding with magnetic octupoles

We simulate the trajectories of magnetic octupole colloids driven by periodic loops of an external magnetic field and placed above a two-dimensional threefold symmetric magnetic pattern. The octupoles avoid the threefold symmetric points above the lattice, either in a topologically trivial way, or by nontrivially winding around these high symmetry points both with respect to their position and with respect to their orientation. We calculate the full dynamical phase space of this winding behavior by changing both lattice symmetries and modulation loops. We further use the nontrivial topology to braid with octupoles and we supply a protocol for both braiding and weaving with such microscopic particles. Our classical external field command should work equally well for quantum mechanical octupoles on magnetic nanopatterns, providing an explicit protocol for the exchange of anyonic quantum particles.