PACS 73.23.Ra, 73.63.Fg The carbon tori and nanotubes, either pure or hole-doped, are examined. In the two achiral (zigzag and armchair) systems the momentum lines are parallel to the edges of the Brillouin zone or the3 03 eV is the overlap integral for graphene) contour. The amplitude and shape of the persistent currents in these cases is predicted by an analysis of the geometry of the reciprocal space, and confirmed by a numerical simulation. Special attention is paid to the differences between the currents in straight nanotube and nanotorus topologies, and other possible configurations of chirality and doping at which the persistent currents may be enhanced are hinted at.