  The Chern numbers for Hofstadter models with rational flux 2*pi*p/q are
partially determined by a Diophantine equation. A Mod q ambiguity remains. The
resolution of this ambiguity is only known for the rectangular lattice with
nearest neighbors hopping where it has the form of a "window condition". We
study a Hofstadter butterfly on the triangular lattice for which the resolution
of ambiguity is open. In the model many pairs (p,q) satisfy a window condition
which is shifted relative to the window of the square model. However, we also
find pairs (p,q) where the Chern numbers do not belong to any contiguous
window. This shows that the rectangular model and the one we study on the
triangular lattice are not adiabatically connected: Many gaps must close. Our
results suggest the conjecture that the mod q ambiguity in the Diophantine
equation generically reduces to a sign ambiguity.
