Knots topological chirality

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

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

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... 

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. 

We can use this same approach to prove that other molecular knots and links are topologically chiral. For example, consider the molecular link illustrated in Figure 18. This catenane was synthesized by Nierengarten et al. [12]. For this molecule the set T(G) consists of many unlinks together with many copies of the (4,2)-torus link, illustrated as L in Figure 12. However we saw earlier that this unoriented link is topologically chiral. Therefore, the molecular (4,2)-torus link is topologically chiral as well. 

The method that we have described enables us to prove the topological chirality of most topologically chiral molecular knots and links. 

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-... 

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