  Recent advances in realizing artificial gauge fields on optical lattices
promise experimental detection of topologically non-trivial energy spectra.
Self-similar fractal energy structures generally known as Hofstadter
butterflies depend sensitively on the geometry of the underlying lattice, as
well as the applied magnetic field. The recent demonstration of an adjustable
lattice geometry [L. Tarruell \textit{et al.}, Nature 483, 302--305 (2012)]
presents a unique opportunity to study this dependence. In this paper, we
calculate the Hofstadter butterflies that can be obtained in such an adjustable
lattice and find three qualitatively different regimes. We show that the
existence of Dirac points at zero magnetic field does not imply the topological
equivalence of spectra at finite field. As the real-space structure evolves
from the checkerboard lattice to the honeycomb lattice, two square lattice
Hofstadter butterflies merge to form a honeycomb lattice butterfly. This
merging is topologically non-trivial, as it is accomplished by sequential
closings of gaps. Ensuing Chern number transfer between the bands can be probed
with the adjustable lattice experiments. We also calculate the Chern numbers of
the gaps for qualitatively different spectra and discuss the evolution of
topological properties with underlying lattice geometry.
