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We present a study of phase transitions of the Curie--Weiss Potts model at(inverse) temperature $\beta$, in presence of an external field $h$. Boththermodynamic and topological aspects of these transitions are considered. Forthe first aspect we complement previous results and give an explicit equationof the thermodynamic transition line in the $\beta$--$h$ plane as well as themagnitude of the jump of the magnetization (for $q \geqslant 3)$. The signatureof the latter aspect is characterized here by the presence or not of a giantcomponent in the clusters of a Fortuin--Kasteleyn type representation of themodel. We give the equation of the Kert\'esz line separating (in the$\beta$--$h$ plane) the two behaviours. As a result, we get that this lineexhibits, as soon as $q \geqslant 3$, a very interesting cusp where itseparates from the thermodynamic transition line.

Thermodynamic versus Topological Phase Transitions: Cusp in the Kert\'esz Line