Hopf oriented

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 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). 

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 unique TEE framework facilitates -conjugation with pendant aromatic substituents by allowing coplanar orientation throughout the molecular core, as was first witnessed in the X-ray crystal structure of tetrakis(phenylethynyl)ethene 1 determined by Hopf and co-workers [9]. In contrast, coplanarity is prevented by steric interactions in molecules such as cis-stilbenes or tetraphenylethene 2 [10] (Figure 2). The planarity makes it possible for 1 to form highly ordered 1 2 stoichiometric donor-acceptor -complexes in the solid state with electron-deficient molecules such as 2,4,7-trinitrofluoren-9-one and (2,4,7-trinitrofluoren-9-ylidene)malonitrile [11]. In solution, relatively weak 1 1 complexes with each of these two acceptors are formed, with association constants of 7.9 m 1 and 31.5 m 1, respectively, at 300 K in CDCl3. 

Figure 5. Bifurcation diagram on the plane of the two control parameters p and a. The solid lines 1 and 2 mark the primary instability, where the homogeneous homeotropic orientation becomes unstable. At 1, the bifurcation is a stationary (pitchfork) bifurcation, and a Hopf one at 2. The two lines connect in the Takens-Bogdanov (TB) point. The solid lines 3 and 4 mark the first gluing bifurcation and the second gluing bifurcation respectively. The dashed lines 2b and 3b mark the lines of the primary Hopf bifurcation and the first gluing bifurcation when calculated without the inclusion of flow in the equations.

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