  The $q$-state Potts model has stood at the frontier of research in
statistical mechanics for many years. In the absence of a closed-form solution,
much of the past efforts have focused on locating its critical manifold,
trajectory in the parameter $\{q, e^J\}$ space where $J$ is the reduced
interaction, along which the free energy is singular. However, except in
isolated cases, antiferromagnetic (AF) models with $J<0$ have been largely
neglected. In this paper we consider the Potts model with AF interactions
focusing on deducing its critical manifold in exact and/or closed-form
expressions.
  We first re-examine the known critical frontiers in light of AF interactions.
For the square lattice we confirm the Potts self-dual point to be the sole
critical point for $J>0$. We also locate its critical frontier for $J<0$ and
find it to coincide with a solvability condition observed by Baxter in 1982.
For the honeycomb lattice we show that the known critical point holds for {all}
$J$, and determine its critical $q_c = \frac 1 2 (3+\sqrt 5) = 2.61803$ beyond
which there is no transition. For the triangular lattice we confirm the known
critical point to hold only for $J>0$.
  More generally we consider the centered-triangle (CT) and Union-Jack (UJ)
lattices consisting of mixed $J$ and $K$ interactions, and deduce critical
manifolds under homogeneity hypotheses. For K=0 the CT lattice is the diced
lattice, and we determine its critical manifold for all $J$ and find $q_c =
3.32472$. For K=0 the UJ lattice is the square lattice and from this we deduce
both the $J>0$ and $J<0$ critical manifolds and find $q_c=3$ for the square
lattice. Our theoretical predictions are compared with recent tensor-based
numerical results and Monte Carlo simulations.
