Slow decompression of crystalline benzene in large-volume high-pressure cells has recently achieved synthesis of a novel one-dimensional allotrope of sp(3) carbon in which stacked columns of benzene molecules rehybridize into an ordered crystal of nanothreads. The progenitor benzene molecules function as six-valent one-dimensional superatoms with multiple binding sites. Here we enumerate their hexavalent bonding geometries, recognizing that the repeat unit of interatomic connectivity ("topological unit cell") need not coincide with the crystallographic unit cell, and identify the most energetically favorable cases. A topological unit cell of one or two benzene rings with at least two bonds interconnecting each adjacent pair of rings, accommodates 50 topologically distinct nanothreads, 15 of which are within 80 meV/carbon atom of the most stable member. Optimization of aperiodic helicity reveals the most stable structures to be chiral. We generalize Euler's rules for ring counting to cover this new form of very thin one-dimensional carbon, calculated their physical properties, and propose a naming convention that can be generalized to handle nanothreads formed from other progenitor molecules.