  We study the effects of frame-dragging on the structure of force-free
magnetospheres around rotating black holes. For slowly rotating black holes,
analytic explorations often focus on transforming an exact force-free solution
applicable to a static black hole into a solution approximately applicable to a
slowly-rotating black hole via perturbations in black hole spin. We show that
the single perturbed monopolar solution most commonly arrived at using such
techniques is in fact a separatrix between two broad classes of solutions:
those with poloidal magnetic fields that bend upwards towards the azimuthal
axis and those that bend downwards towards the equatorial plane. This is
because frame-dragging affects the toroidal magnetic field distributions,
changing the force balance between poloidal magnetic field lines.
