  The classification of loop symmetries in Kitaev's honeycomb lattice model
provides a natural framework to study the abelian topological degeneracy. We
derive a perturbative low-energy effective Hamiltonian, that is valid to all
orders of the expansion and for all possible toroidal configurations. Using
this form we demonstrate at what order the system's topological degeneracy is
lifted by finite size effects and note that in the thermodynamic limit it is
robust to all orders. Further, we demonstrate that the loop symmetries
themselves correspond to the creation, propagation and annihilation of
fermions. Importantly, we note that these fermions, made from pairs of
vortices, can be moved with no additional energy cost.
