  In two dimensions, topological phases of free Majorana fermions coupled to a
$\mathbb{Z}_2$ gauge field are known to be classified according to the Chern
number $\nu \in \mathbb{Z}$. Its value mod 16 specifies the type of anyonic
excitations. In this paper, we investigate triangular vortex configurations
(and their dual) in the Kitaev honeycomb model and show that fourteen of these
sixteen phases can be obtained by adding a time-reversal symmetry-breaking
term. Missing phases are $\nu=\pm 7$. More generally, we prove that any
periodic vortex configuration with an odd number of vortices per geometric unit
cell can only host even Chern numbers whereas odd Chern numbers can be found in
other cases.
