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

We discuss a new class of identities between correlation functions which ar ise from a local Zz invariance of the partition function of the q-state Pot ts model on general graphs or lattices. Their common feature is to relate t he thermal operators of the Potts model to some topological properties of t he Fortuin-Kasteleyn clusters. In particular, it turns out that any even co rrelation function can be expressed in terms of observables which probe the linking properties of these clusters. This generalizes a class of analogou s relations recently found in the Ising model.