  We consider an extension of the Kitaev honeycomb model based on arbitrary
dimer coverings satisfying matching rules. We focus on three different dimer
coverings having the smallest unit cells for which we calculate the
ground-state phase diagram. We also study one- and two-vortex properties for
these coverings in the Abelian phases and show that vortex-vortex interactions
can be attractive or repulsive. These qualitative differences are confirmed
analytically by high-order perturbative expansions around the isolated-dimer
limit. Similarities and differences with the original Kitaev honeycomb model
are discussed.
