  We compute the critical polymials for the q-state Potts model on all
Archimedean lattices, using a parallel implementation of the algorithm of
(Jacobsen, 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 $6 \times 6$ unit cells, and the root in the temperature variable
$v=e^K-1$ is determined numerically at $q=1$ for bases of size $8 \times 8$.
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.
