Torus without latitudes

Until now, all of the toroidal systems we have described and constructed can be classified according to the arrangement of pentagons and hexagons in different positions of lattitudes. In other word, the latitudes are well defined. Is there any toroidal system not in this family? The anwser is yes. The beaded molecule shown in the following figure (created by Chuang) is an example for toroids without latitudes (discovered by Ihara). Through the Goldberg transformation, a whole family of compounds can be generated. In particular, the five arms of the compounds can be elongated by inserting more hexagons in them, we can thus obtain the armchair-typed toroidal carbon nanotubes.
Unfortunately, I don't know if it is possible to generalize our classification scheme to include this type of compounds yet.