Achirality topological chirality

We shall now see how to apply the theorem to the molecular trefoil knot, which was illustrated in Figure 17. We can create a molecular cell complex G by replacing each isolated benzene ring by a cell and each chain of three fused rings by a single cell. We prove by contradiction that our molecular cell complex is topologically chiral. Suppose that it is topologically achiral. Then there is a defor-... 

Amide-linked catenanes and rotaxanes can play a major role in the study of rare forms of chirality, e.g. topological chirality and cycloenantiomerism [60]. Resolution of enantiomeric catenanes, rotaxanes, and pretzelanes has been achieved by HPLC on chiral column materials, but further work must be performed to determine absolute configurations and to realize new chiral skeletons composed of achiral building blocks. Topological asymmetric synthesis still belongs to dreams of the future yet should be kept in mind. 

An object, such as a knot, or a link, or a graph, is topologically chiral if and only if it cannot be converted into its mirror image by continuous deformation (ambient isotopy) in the object s space otherwise it is topologically achiral. 

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. 

The knots in Figure 21 are all prime knots because they cannot be divided (factored) into smaller, nontrivial knots. Prime knots are the building blocks of composite knots and of links. Like prime numbers, which yield composite numbers upon multiplication, or like atoms in chemistry, which yield molecules upon combination, prime knots are the elementary units of knot theory. Composite knots are exemplified by the topologically achiral square knot and the topologically chiral granny knot (Figure 22). In each of these knots, a plane perpendicular to the... 

The existence of topological rubber gloves proves that the absence of rigidly achiral presentations is a necessary but not a sufficient condition for topological chirality. 

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

With few exceptions, all nonoriented chemical links reported in the literature are topologically achiral. One of the rare exceptions is the two-component 4-crossing link 4 (Figure 31).114 The molecule in the depicted conformation, and indeed in any imaginable conformation, is geometrically chiral. It is therefore reasonable to conjecture that it is also topologically chiral. Proof that 4 is indeed topologically chiral was provided after the development of suitable polynomials.115... 

While orientation normally imparts topological chirality, as in the case of the Hopf link, this is not always the case. For example, the Borromean link remains amphicheiral even after orientation.115- 119 This is easily demonstrated with reference to the 5), presentation of the nonoriented link [Figure 27(b)] No matter in which direction the three rings are oriented, the resulting diagram remains centrosymmetric and therefore rigidly achiral. 

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... 

In stark contrast to the numerous functions that are available to measure geometrical chirality, no measure has yet been reported for the quantification of topological chirality. In analogy to geometrical chirality measures, topological chirality measures %(K) must satisfy two minimal conditions They can be equal to zero if and only if the knot or link is achiral, and they have to have the same absolute value for two topological enantiomorphs. 

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