  Motivated by the discovery of quantum anomalous Hall effect in Cr-doped
$\text{Bi}_{2}\text{Se}_{3}$, we study the generic states for magnetic
topological insulators and explore the physical properties for both magnetism
and itinerant electrons. First-principles calculations are exploited to
investigate the magnetic interactions between magnetic Co atoms adsorbed on the
$\text{Bi}_{2}\text{Se}_{3}$ (111) surface. Due to the absence of inversion
symmetry on the surface, there are Dzyaloshinskii-Moriya like twisted spin
interactions between the local moments of Co ions. These non-ferromagnetic
interactions twist the collinear spin configuration of the ferromagnet and
generate various magnetic orders beyond a simple ferromagnet. Among them, the
spin spiral state generates alternating counter-propagating modes across each
period of spin states, and the skyrmion lattice even supports a chiral mode
around the core of each skyrmion. The skyrmion lattice opens a gap at the
surface Dirac point, resulting in anomalous Hall effect. These results would
inspire further experimental investigation of magnetic topological insulators.
