Trigonometrical sums connected with the chiral Potts model, Verlinde dimension formula, two-dimensional resistor network, and number theory

\ \ We have recently developed methods for obtaining exact two-point resistance of the complete graph minus $N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving the Scott's conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of $SU(2)$ and $SO(3)$ WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, $2\times N$. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory.