Thermal operators and cluster topology in the q-state Potts model

We discuss a new class of identities between correlation functions which arise from a local Zz 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 generalizes a class of analogous relations recently found in the Ising model.