T = 0 partition functions for Potts antiferromagnets on square lattice strips with (twisted) periodic boundary conditions

We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (or, equivalently, the chromatic polynomial) for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and (i) periodic and (ii) twisted periodic boundary conditions in the longitudinal direction, so that the strip graphs are embedded on (i) a torus and (ii) a Klein bottle. In the limit of infinite length, we calculate the exponent of the entropy, W(q), show it to be the same for (i) and (ii), and determine its analytic structure.