  We calculate analytically the Bogoliubov excitation spectrum of a toroidal
spin-1 Bose-Einstein condensate that is subjected to a homogeneous magnetic
field and contains vortices with arbitrary winding numbers in the $m_F=\pm 1$
components of the hyperfine spin. We show that a rotonlike spectrum can be
obtained, or an initially stable condensate can be made unstable by adjusting
the magnitude of the magnetic field or the trapping frequencies. The structure
of the instabilities can be analyzed by measuring the particle densities of the
spin components. We confirm the validity of the analytical calculations by
numerical simulations.
