  We show an exact equivalence of the free energy of the $q$-state Potts
antiferromagnet on a lattice $\Lambda$ for the full temperature interval $0 \le
T \le \infty$ and the free energy of the $q$-state Potts model on the dual
lattice for a semi-infinite interval of complex temperatures (CT). This implies
the existence of two quite different types of CT singularities: the generic
kind, which does not obey universality or various scaling relations, and a
special kind which does obey such properties and encodes information of direct
physical relevance. We apply this observation to characterize CT properties of
the Potts model on several lattices, to rule out two existing conjectures, and
to determine the critical value of $q$ above which the Potts antiferromagnet on
the diced lattice has no phase transition.
