The-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.

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Carbon nanotubes are very promising new materials that fit perfectly well into the miniaturisation of technology, but at the same time show a fascinating quantum structure. When the tube is closed to form a torus the structure acquires discrete molecular characteristics. In the present paper the electronic levels of tori are obtained by a double folding of the Brillouin zone of graphitic carbon. An illustrative example of an hypothetical giant torus with molecular symmetry is fully analysed, with special emphasis on its pictorial representation.