Hopf link

We shall use this result to enable us to compute the P-polynomial of the oriented link, illustrated as L in Figure 5. This link is known as the Hopf link. We choose the upper crossing to change so that we have L+ and Lq as indicated in the figure. 

Notice that the orientation of a link may affect its polynomial. For example, the Hopf link in Figure 6 has one component which is oriented in a different di-... 

Figure 5. We compute the P-polynomial of the oriented Hopf link which is represented by L. ...

Figure 6. A Hopf link with a different orientation from that of L in Figure 5.

In order to see how to apply this theorem we can consider the oriented Hopf link which was illustrated in Figure 5. We determined above that the P-polynomial of this oriented Hopf link L is P(L) = Z3m 1 + lm l - Im. If we interchange l and r1 we will obtain the polynomial P(JL) = / 3m-1 + - rlm. Since P(L) ... 

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

Figure 8. An oriented Hopf link is topologically chiral.

Figure 9. By turning over one component we obtain the mirror image of the unoriented Hopf link.

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

In a similar way we can prove that the embedded cell complex of the molecular (4,2)-torus link (see Figure 18) is topologically chiral. Also, by adding appropriate labels we can similarly prove the topological chirality of the oriented embedded cell complex of the molecular Hopf link (see Figure 19). 

Figure 26. (a) Minimal diagram of the Hopf link, 2 . (h) A [2]catenane105 abstractly represented by... 

Like knots, links may be prime or composite. The Hopf and Borromean links are examples of prime links because they cannot be divided (factored) into smaller, nontrivial links. Figure 30(a) is the minimal diagram of a composite link that is the abstract representative of some [3]catenanes, one of which is depicted in Figure 30(b).103b That the three-component link is a composite link is shown by the fact that a plane perpendicular to the plane of projection (dashed line) and pierced in exactly two points cuts the link in half If the open ends on both sides of the plane are now joined to form closed curves, two Hopf links result. In analogy to composite knots, the three-component link in Figure 30(a) is denoted by 2 2j, and the five-component composite link that represents olympiadane by 2 2 2 2. ... 

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. 

Figure 1.3.9 Three examples of self-catenated nets with two catenated rings (Hopf links) in black 6-rings (rob), 8-rings (coe), 12-rings (twt).

The main feature of polycatenation is that the whole catenated array has a higher dimensionality then that of each of the component motifs. These motifs can, in principle, be OD, ID or 2D species that must contain closed loops and that are interlocked, as for interpenetrating nets, via topological Hopf links [31] (see Fig. 1.3.11). Each motif can be catenated with a finite or also with an infinite number of other independent motifs but not with all (see Fig. 1.3.10). 

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