Generalized classification scheme of toroidal and helical carbon nanotubes

In this study, we develop a generalized classification scheme for toroidal (TCNT) and helical carbon nanotubes (HCNT) containing both pentagons and heptagons simultaneously. We show that a particular class of TCNTs with n-fold rotational symmetry and well-defined latitude coordinates can be uniquely characterized by a set of four indices, and each of the indices can be linked to the relative arrangement of pentagons and heptagons in the corresponding torus. Chiral isomers or the corresponding helical derivatives, HCNTs, can also be readily derived either by introducing a chiral vector or dissecting a distorted TCNT through certain longitude. This generalized scheme can generate the whole family of fullerenes with genus one, ranging from giant TCNTs down to small ones containing only a few hundred atoms. To the best of our knowledge, almost all of the construction methods for TCNTs in the literature belong to special cases of our generalized classification scheme.