Orientational automorphism

Proot The universal cover of such a polycycle is an (r, r)-polycycle with a nontrivial group of fixed-point-free automorphisms. The list of (r, < )-polycycles for r,q < 4 is known (see Section 4.2). Inspection of this list gives only the infinite polycycles Prisntoo = P2X Pz and APrisrrioo = Tr%. Their orientable quotients are the infinite series of prisms Prismm (m > 2), the infinite series of antiprisms APrisnim... 

Proof. Hike such a polycycle. Its universal cover is an elementary (5,3)-polycycle, whose automorphism group contains some fixed-point-free transformation. Inspection of the list of elementary (5,3)-polycycles in Figure 7.2 yields only Eg = snub Prismoo as a possibility. Snub Prismm is obtained from the group of translations by m faces and the non-orientable quotients if the group contains also some translation followed by reflection. ... 

Let us now determine all elementary ( 2,3,4,5), 3)gOT-polycycles. Hie universal cover P of such a polycycle P is an elementary ( 2,3,4,5, 3)-polycycle, whichhas a non-trivial fixed-point-free automorphism group in Aut(P). Consideration of the above list of polycycles yields snub Prisma, as the only possibility. The polycycles snub Prismm and its non-orientable quotients arise in this process. ... 

If P is an elementary ( 2,3], 5)ge -polycycle, which is not a ( 2,3, 5)-polycycle, then its universal cover P is an elementary ( 2,3, 5)-polycycle, which has a fixed-point-free automorphism group included in Aut(P). Clearly, only snub APrismoo is such and it yields the infinite series of snub APrisnim and its non-orientable quotients. ... 

Remark (A test on chirality) Orientation functions obtained from each other by a graph-automorphic renumbering are isomorphic. A conformation is achiral if and only if its orientation function is isomorphic to its negative. 

POFs Xo nd Xi correspond to the two enantiomers shown in Figure 4.3. POF X2 is not chemically realizable. However, it is geometrically realizable Any planar placement of the five atoms has the POF X2-b There are 3 = 729 abstract POFs for structure 4.2b, cyclohexane. Each of the 6 selected increasing quadruples can have the orientation plus, minus, or zero independently. However, cyclohexane has 12 graph automorphisms. Considering relabelings according to these permutations, some of the abstract POFs are isomor-... 

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