  We present exact calculations of the zero-temperature partition function for
the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial)
for families of arbitrarily long strip graphs of the square and triangular
lattices with width $L_y=4$ and boundary conditions that are doubly periodic or
doubly periodic with reversed orientation (i.e. of torus or Klein bottle type).
These boundary conditions have the advantage of removing edge effects. In the
limit of infinite length, we calculate the exponent of the entropy, $W(q)$ and
determine the continuous locus ${\cal B}$ where it is singular. We also give
results for toroidal strips involving ``crossing subgraphs''; these make
possible a unified treatment of torus and Klein bottle boundary conditions and
enable us to prove that for a given strip, the locus ${\cal B}$ is the same for
these boundary conditions.
