  We report several results concerning $W(\Lambda,q)=\exp(S_0/k_B)$, the
exponent of the ground state entropy of the Potts antiferromagnet on a lattice
$\Lambda$. First, we improve our previous rigorous lower bound on $W(hc,q)$ for
the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to
the first eleven terms with the large-$q$ series for $W(hc,q)$. Second, we
investigate the heteropolygonal Archimedean $4 \cdot 8^2$ lattice, derive a
rigorous lower bound, on $W(4 \cdot 8^2,q)$, and calculate the large-$q$ series
for this function to $O(y^{12})$ where $y=1/(q-1)$. Remarkably, these agree
exactly to all thirteen terms calculated. We also report Monte Carlo
measurements, and find that these are very close to our lower bound and series.
Third, we study the effect of non-nearest-neighbor couplings, focusing on the
square lattice with next-nearest-neighbor bonds.
