  We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent
graphs, on a lattice. We suggest some routes by which such a Hilbert space
could be the low-energy subspace of a model of quantum spins on a lattice with
short-ranged interactions. We then explain conditions which a Hamiltonian
acting on this string net Hilbert space must satisfy in order for its ground
state and low-lying quasiparticle excitations to be in the DFib topological
phase. Using the string net wavefunction, we describe the properties of this
phase. Our discussion is informed by mappings of string net wavefunctions to
the chromatic polynomial and the Potts model.
