Comments on structural types of toroidal carbon nanotubes

We clarify the relationships between the eight structural types of toroidal carbon nanotubes (TCNTs), which can be identified as the eight corners of a cube of structural transformation. The four families with Dnh symmetry can be related by rim rotations, and the same is true for those with Dnd symmetries. These two sets are then connected by horizontal shiftings, thereby completing the cube. Moreover, we further point out that there are five more highly symmetric Dnh structural types that can be derived from performing the generalized Stone-Wales transformatoin on certain TCNTs with Dnh structural types.