Torus link

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. 

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. 

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). 

There is another viewpoint on the torus links. The two strands in this case could be treated as the edges of a ribbon. Therefore, the topological theory of torus links is actually the theory of ribbons. 

As a final example we consider noncovalent molecular complex formation with the macrocyclic ligand a-cyclodextrin, a natural product consisting of six a-D-glucose units linked 1-4 to form a torus whose cavity is capable of including molecules the size of an aromatic ring. Table 4-3 gives some rate constants for this reaction, where L represents the cyclodextrin and S is the substrate ... 

A torus, which is the result of linking opposite sides of a two-dimensional rectangular SOM. 

CDs are torus-shaped cyclic molecules of a-(l —>4)-linked glucose, with 6, 7, or 8 glucose residues in the ring or y-cyclodextrins, respectively). They are... 

CDs are by far the most popnlar chiral selectors in CE and will therefore be discussed in more detail than the others. CDs are torus-shaped, cyclic, a(f,4)-linked oligomers of D( + )-glucopyranose. They contain between six and 12 D(-l-)-glucopyranose units, bnt only those with 6 (a-CD), 7 ( -CD) and 8 (/-CD) units are currently used in chiral separations (Figure 1). The interior of the CD cavity is relatively hydrophobic, while the outside rims are more hydrophilic. The rim of the wider side of the CD contains chiral secondary hydroxyl groups, while achiral primary hydroxyl groups occupy the opposite smaller opening. 

The pneumatic drying model was solved numerically for the drying processes of sand particles. The numerical procedure includes discretization of the calculation domain into torus-shaped final volumes, and solving the model equations by implementation of the semi-implicit method for pressure-linked equations (SIMPLE) algorithm [16]. The numerical procedure also implemented the Interphase Slip Algorithm (IPSA) of [17] in order to account the various coupling between the phases. The simulation stopped when the moisture content of a particle falls to a predefined value or when the flow reaches the exit of the pneumatic dryer. 

Equations (3.33) and (3.34) make a succinct statement about quantum-classical correspondence. Specilically, in the classical limit, stationary quantum eigendistributions correspond to stationary classical eigendistributions at the related quantized action. Secondly, nonstationary quantum eigendistributions pD>m(Ij,0) correspond to nonstationary classical eigendistributions. These consist of a nonuniform distribution on a torus at intermediate action I(m+D)/2 with the nonuniformity determined by the difference (n — m) times the angles on this torus.59 Thus, from this viewpoint there is a direct link between eigendistributions in quantum and classical mechanics in the h - 0 limit. 

The mouth of the torus-shaped cyclodextrin molecule has a larger circumference than at the base and is linked to secondary hydroxyl groups of the Cj and C3 atoms of each glucose unit (see figure 5.8). 

The trapping process was simulated using a torus centered around each repeat unit in the cyclic.Any empty torus was considered a pathway for a chain of specified diameter to thread and then incarcerate the cyclic once the end-linking process has been completed. Simulations were consistent with experimental trapping efficiencies. It is possible to interpret these experimental results in terms of a power law for the trapping probabilities and fractal cross sections for the PDMS chains. 

A different approach has been adopted by Morton and Grishanov (2009) where two auxiUary closed curves related to the torus were included into torus diagrams for 2-structures, thus producing a multicomponent link. The... 

Cycloamyloses (cyclic a-l,4-linked oligomers of D-glucose) have a toroidal or doughnuf -shaped structure. The primary hydroxy groups are located on one side of the torus while the secondary ones lie on the other side. Relative to water the interior of the cycloamylose torus is apolar. The catalytic properties of cycloamyloses depend on the formation of inclusion complexes with the substrate and subsequent catalysis by either the hydroxy, or other groups, located around the circumference of the cavity (Komiyama and Bender, 1984 Page and Crombie, 1984). 

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