Chirality topological

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

Out of many exciting recently obtained topologically distinct structures combined rotaxanes 41,400 [23], pretzel-molecule 35 [24a] and bis(pretzelane) 401 [24b], a topologically chiral [2]catenane [25] and an interestiig catenated... 

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

P(L), the theorem tells us that the oriented Hopf link is topologically chiral. This... 

Figure 8. An oriented Hopf link is topologically chiral.

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 can also use link polynomials to prove that certain unoriented links are topologically chiral. For example, let L denote the (4,2)-torus link which is illustrated on the left in Figure 12. This is called a torus link because it can be embedded on a torus (i.e. the surface of a doughnut) without any self-intersections. It is a (4,2)-torus link, because, when it lies on the torus, it twists four times around the torus in one direction, while wrapping two times around the torus the other way. Let L denote the oriented link that we get by putting an arbitrary orientation on each component of the (4,2)-torus link, for example, as we have done in Figure 12. Now the P-polynomial of L is P(L ) = r5m l - r3m x + ml 5 -m3r + 3m r3. 

Theorem. Let G be an embedded graph. If there is an element of T(G) which is topologically chiral and which cannot be deformed to the mirror image of any other element of T(G), then G is topologically chiral [9]. 

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. 

Figure 20. The labeling of this Hopf link gives it an orientation, which makes it topologically chiral.

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