  We study a simple magnetic system composed of periodically modulated magnetic
dipoles with an easy axis. Upon adjusting the modulation amplitude alone,
chains and two-dimensional stacked chains exhibit a rich magnon spectrum where
frequency gaps and magnon speeds are easily manipulable. The blend of
anisotropy due to dipolar interactions between magnets and geometrical
modulation induces a magnetic phase with fractional Zak number in infinite
chains and end states in open one-dimensional systems. In two dimensions it
gives rise to topological modes at the edges of stripes. Tuning the amplitude
in two-dimensional lattices causes a band touching, which triggers the exchange
of the Chern numbers of the volume bands and switches the sign of the thermal
conductivity.
