Examples of Graph—Theoretically Nonplanar Polyhexes

Already Balaban (1969) mentioned (in a footnote) an interesting polyhex structure, which later has been designated figure-eight (Randic, Nikolid and Trinajstic 1988 Balaban 1988) see Fig. 6. Randid, Nikolid and Trinajstic (1988) characterized this structure as A twisted loop overlapping without intersection. They called it cyclohelicene, and indeed it can be interpreted as two hexahelicenes fused to each other in a macrocycle. The mentioned authors reported K — 260 as the Kekuld structure count for cyclohelicene (Fig. 6). One year later, Randid, 

Gimarc et al. (1989) offered a more detailed theoretical study of the same structure. 

The poly hex under consideration (Fig. 6) is embeddable in the hexagonal lattice, and it has overlapping edges or, more precisely, two overlapping hexagons. Hence it is reasonable to consider this system as helicenic, and that has actually been done previously (Randid, Nikolid and Trinajstid 1988 Randic, Gimarc et al. 1989 Cyvin BN, Brunvoll and Cyvin 1992b). However, cyclohelicene is not a corohelicene, and of course not a coronoid. It is true that two perimeters are present, but they are symmetrically equivalent none of them qualifies to be identified especially as the outer— or inner perimeter. There are two enantiomers of the structure in question (Fig. 6). 

The designation [12]cyclohelicene (Randid, Gimarc et al. 1989) is ambiguous. Another figure-eight isomer of C gH2 is depicted in Fig. 7. It was put forward by Balaban (1988) and is perhaps even more interesting than the structure of Fig. 6. The two C gH2 isomers under 

Balaban (1969 1988) depicted actually the two enantiomorphic forms of a [14]cyclo— helicene (rather than [12]cyclohelicene) and augmented it to a system with 22 hexagons and two loops  

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