[en] We compute critical polynomials for the q-state Potts model on the Archimedean lattices, using a parallel implementation of the algorithm of Jacobsen (2014 J. Phys. A: Math. Theor 47 135001) that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size 6 × 6 unit cells, and the root in the temperature variable $v={\mathrm{e}}^K-<!--\; ???\; -->1$ is determined numerically at q = 1 for bases of size 8 × 8. This leads to improved results for bond percolation thresholds, and for the Potts-model critical manifolds in the real (q, v) plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the $(3,{12}^2)$ lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker–Kadanoff phase for the lattices studied. (paper)