  We study azimuthal spin-wave (SW) excitations in a circular ferromagnetic
nanodot in different inhomogeneous, topologically non-trivial magnetization
states, specifically, vortex, Bloch-type skyrmion and N\'eel-type skyrmion
states. Continuous mapping of the SW spectrum between these states is realized
with gradual change of the out-of-plane magnetic anisotropy and
Dzyaloshinskii-Moriya exchange interaction (DMI). Our study shows lifting of
the SW frequencies degeneracy and change in systematics of the frequency
levels. The change is induced by a geometrical Berry phase, that is present for
the dot-edge localized SWs in a vortex state and vanishes in skyrmion states.
Furthermore, channeling of the azimuthal SWs localized at the skyrmion edge is
present and induces large frequency splitting. This is attributed to DMI
induced nonreciprocity, while coupling of the breathing and gyrotropic modes is
related to soliton motion. Finally, an efficient coupling of the dynamic
magnetization in the skyrmion state to uniform magnetic field in nanodots with
non-circular symmetry is shown.
