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

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