  We consider the topological theories of cond-mat/0404617 and cond-mat/0610583
and study ground state amplitudes of string net configurations which consist of
large chunks $G$ of (trivalent) regular lattice. We evaluate these amplitudes
in two different ways: first we use the Turaev-Viro prescription to write the
amplitude as a sum over labelings of the faces of $G$, and second we use the
local rules that constrain the amplitude (the $F$-matrix) to resolve subgraphs
in creative ways. In the case of the Doubled Fibonacci theory this second way
allows us to produce loop models. In particular, we show that the hard hexagon
model is equivalent to an anisotropic loop model. Many other interesting
equivalences can presumably be obtained.
