  Spin-orbit coupling has novel spin-flip symmetries, a spin-1 spinor
Bose-Einstein condensate owns meaningful interactions, and a toroidal trap is
topologically nontrivial. We incorporate the three together and study the
ground-state phase diagram in a Rashba spin-orbit-coupled spin-1 Bose-Einstein
condensate with a toroidal trap. The spin-flip symmetries give rise to two
different interesting phases: persistent flows with a unit phase winding
difference between three components, and necklace states with even
petal-number. The existing parameter regimes and properties of these phases are
characterized by two-dimension numerical calculations and an azimuthal
analytical one-dimension model.
