  We explore the interplay between topology and eigenmodes by changing the
stabilizing mechanism of skyrmion lattices (skX). We focus on two prototypical
ultrathin films hosting an hexagonal (Pd/Fe/Ir(111)) and a square (Fe/Ir(111))
skyrmion lattice, which can both be described by an extended Heisenberg
Hamiltonian. We first examine whether the Dzyaloshinkskii-Moriya, or the
exchange interaction as the leading energy term affects the modes of the
hexagonal skX of Pd/Fe/Ir(111). In all cases, we find that the lowest frequency
modes correspond to internal degrees of freedom of individual skyrmions, and
suggest a classification based on azimuthal and radial numbers $(l,p)$, with up
to $l=6$, and $p=2$. We also show that the gyration behavior induced by an
in-plane field corresponds to the excitation of $l=1$ deformation modes with
varying radial numbers. Second, we examine the square lattice of skyrmions of
Fe/Ir(111). Its stabilization mechanism is dominated by the 4-spin interaction.
After relaxation, the unit cell does not carry a topological charge, and the
eigenmodes do not correspond to internal skyrmion deformations. By reducing the
4-spin interaction, the integer topological charge is recovered, but the charge
carriers do not possess internal degrees of freedom, nor are they separated by
energy barriers. We conclude that a 4-spin dominated Hamiltonian does not yield
skyrmion lattice solutions, and that therefore, a nontrivial topology does not
imply the existence of skyrmions.
