  We report the results of a theoretical investigation of the stability of a
toroidal vortex bound by an interface. Two distinct instability mechanisms are
identified that rely on, respectively, surface tension and fluid inertia,
either of which may prompt the transformation from a circular to a polygonal
torus. Our results are discussed in the context of three experiments, a
toroidal vortex ring, the hydraulic jump, and the hydraulic bump.
