  Using a sum rule approach we calculate the frequency shifts of the quadrupole
oscillations of a harmonically trapped Bose gas due to the presence of a
quantized vortex. Analytic results are obtained for positive scattering lengths
and large N where the shift relative to excitations of opposite angular
momentum is found to be proportional to the quantum circulation of the vortex
and to decrease as N^{-2/5}. Results are also given for smaller values of N
covering the transition between the ideal gas and the Thomas-Fermi limit. For
negative scattering lengths we predict a macroscopic instability of the vortex.
The splitting of the collective frequencies in toroidal configurations is also
discussed.
