Chiral knots

This molecule has no chiral carbons, nor does it have a rigid shape, but it too has neither a plane nor an alternating axis of symmetry. Compound 32 has been synthesized and has, in fact, been shown to be chiral. Rings containing 50 or more members should be able to exist as knots (33, and see 37 on p. 114 in Chapter 3). Such a knot would be nonsuperimposable on its mirror image. Calixarenes, ° crown ethers, catenanes, and rotaxanes (see p. 113) can also be chiral if suitably substituted. For example, A and B are nonsuperimposable mirror images. 

For a review of chirality in Mobius-strip molecules catenanes, and knots, see Walba, D.M. Tetrahedron, 1985, 41, 3161. 

Chambron, J.-C., Dietrich-Buchecker, Ch., and Sauvage, J.-P. From Classical Chirality to Topologically Chiral Catenands and Knots. 165, 131-162 (1993). 

Some racemates (Figure 3.23) are more efficiently resolved on the bonded-type CSP than the coated-type CSP by using chloroform as a component of the eluent. On the bonded-type CSP of 24n, topologically interesting catenanes and molecular knots are successfully resolved using a hexane-chloroform-2-propanol mixture.185 The first direct HPLC resolution of the smallest chiral... 

From Classical Chirality to Topologically Chiral Catenands and Knots... 

There are >106 knots for n = 16 [23] and by extrapolation there will be >107 for n=18. Thus a ring of -300 carbon atoms has the potential for tens of millions of knotted isomers, most of which will be chiral. The laboratory tests of these conjectures are left as an exercise for the reader. 

Note that the theorem does not detect all topologically chiral knots and oriented links, because there are topologically chiral knots and oriented links whose P-polynomials are nonetheless symmetric with respect to / and l"1. For example, consider the knot which is illustrated in Figure 11. This knot is known by knot theorists as 942 because this is the forty second knot with 9 crossings listed in the standard knot tables (see the tables in Rolfsen s book [9]). Using a computer program we find that the P-polynomial of the knot 942 is P(942) = (-21 2 - 3-212) + m2 l 2 + 4 + l2) - m. Observe that this polynomial is symmetric with respect... 

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