Topological chirality molecular graphs

We shall now apply the techniques that we described above to prove the topological chirality of some molecular knots and links. Note that if we succeed in proving that a molecular graph is topologically chiral then it will follow that the molecule that it represents is chemically chiral, since any molecular motion corresponds to a rigid or flexible deformation of the molecular graph. In particular, it is not chemically possible for one molecular bond to pass through another molecular bond. 

After synthesizing the 3-rung Mobius ladder, Walba conjectured that its molecular bond graph was topologically chiral [16]. In contrast with his conjecture, he... 

In short, the absence of rigidly achiral presentations, planar or otherwise, is a necessary condition for the topological chirality of a molecular graph. Yet, as we shall later see, it is still not a sufficient one. 

Figure 16. Some topologically chiral molecular graphs, (a) The Simmons-Paquette molecule.74 (b) Walba s 3-rung Mobius ladder molecule.75 (c) [4](l,l )[4](3,3 )[3](4 4 )-Ferrocenophan-16-one.76 (d) Triple-layered naphthalenophane.78 Unlabeled vertices represent carbon atoms, and hydrogen atoms are suppressed for clarity. Only one enantiomer of each molecule is shown.

There is no problem in identifying the vertex set in a molecular graph that represents the constitutional formula of a molecule because each vertex bears a one-to-one correspondence to an appropriately labeled atom in the molecule. The relationship of edges in the graph to bonds in the molecule is, however, far less well defined. This point warrants a reiteration of earlier remarks on this theme52,71, 72-79 93 because it has an important bearing on the subject of topological chirality in molecules. 

The topological chirality or achirality of a molecule thus refers exclusively to its molecular graph, and not necessarily to a physically realistic model. Yet, as remarked above, the concept of chirality in chemistry has a well-defined meaning... 

Another topologically chiral K5 molecule is the chiral ferrocenophane ketone derivative 51 (Fig. 8), which is an intermediate in the synthesis of tetrabridged ferrocenophanes [79]. It is important to notice that in order to obtain a non-planar molecular graph, at least four of the ten Fe —C bonds must be taken into account. The K5 subgraph (52) also contains an oriented edge ( —CH2CH2CH2CO-fragment) and a coloured vertex (Fe atom). 

This theorem does not distinguish between chiral polyhedra (as is evident since the molecular graph G does not). But this is well handled if one instead pays attention to the embedding of G in the surface S (presently homeomorphic to a sphere), all as is done by the topological equivalence of Section 9.2.1. Then"" ... 

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