The pi-orbital electronic structure of polyhex carbon tori constructed from bent individual single-wall nanotubes is investigated within the tight-binding approach. Analytical solutions for tori from nanotubes of arbitrary radius, length, chirality, and twisting angle are derived using simple geometrical and band-structure arguments. Vanishing of the gap between highest occupied and lowest unoccupied molecular orbitals for a torus imposes divisibility by 3 on the indices of chiral and twisting vectors, which translates into one graph-theoretical condition: a metallic polyhex torus is constructible as a leapfrog transformation of a smaller polyhex torus. (C) 2000 American Institute of Physics. [S0021-9606(00)70408-9].