New Exact Results for the Potts Model

We present new exact results for the checkerboard $q$-state Potts model with a magnetic field. An exact expression for the partition function is given on some disorder varieties. We argue that the partition function exhibits, even in the presence of a magnetic field, an unexpected $S_4$ symmetry. A universal and exact expression for the magnetization discontinuity at the first-order phasetransition point ($q>\mathrm{}4\mathrm{}$) is proposed. This expression is shown to depend only on $q$ and to be independent of the coupling constants.