  We consider Potts model hypersurfaces defined by the multivariate Tutte
polynomial of graphs (Potts model partition function). We focus on the behavior
of the number of points over finite fields for these hypersurfaces, in
comparison with the graph hypersurfaces of perturbative quantum field theory
defined by the Kirchhoff graph polynomial. We give a very simple example of the
failure of the "fibration condition" in the dependence of the Grothendieck
class on the number of spin states and of the polynomial countability condition
for these Potts model hypersurfaces. We then show that a period computation,
formally similar to the parametric Feynman integrals of quantum field theory,
arises by considering certain thermodynamic averages. One can show that these
evaluate to combinations of multiple zeta values for Potts models on polygon
polymer chains, while silicate tetrahedral chains provide a candidate for a
possible occurrence of non-mixed Tate periods.
