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We present a study of phase transitions of the Curie--Weiss Potts model at (inverse) temperature β, in presence of an external field h. Both thermodynamic and topological aspects of these transitions are considered. For the first aspect we complement previous results and give an explicit equation of the thermodynamic transition line in the β--h plane as well as the magnitude of the jump of the magnetization (for q ≥ 3). The signature of the latter aspect is characterized here by the presence ordoi:10.1209/0295-5075/82/50003 fatcat:srgwhltoyfbctpotwvbbjyj65y