Possible global minimum lattice configurations for Thomson's problem of charges on a sphere

What configuration of N point charges on a conducting sphere minimizes the Coulombic energy? J.J. Thomson posed this question in 1904. For N less than or equal to 112, numerical methods have found apparent global minimum-energy configurations; but the number of local minima appears to grow exponentially with N, making many such methods impractical. Here we describe a topological/numerical procedure that we believe gives the global energy minimum lattice configuration for N of the form N = 10(m(2) + n(2) + mn) + 2 (m, n positive integers). For those N with more than one lattice, we give a rule to choose the minimum one.