  We discuss a new class of identities between correlation functions which
arise from a local Z_2 invariance of the partition function of the q-state
Potts model on general graphs or lattices. Their common feature is to relate
the thermal operators of the Potts model to some topological properties of the
Fortuin-Kasteleyn clusters. In particular it turns out that any even
correlation function can be expressed in terms of observables which probe the
linking properties of these clusters. This generalises a class of analogous
relations recently found in the Ising model.
