Topological chirality oriented links

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. 

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. 

FIGURE 13 Enantiomorphs of topologically chiral constructions. Top Trefoil knot. Center Four-crossing two-component link. Bottom Oriented two-crossing link. 

Figure 3.3 The simplest chiral knots, the left-handed and right-handed trefoil knots T. and T+ no motion of the rope can convert a chiral knot into its mirror image. An orientation is specified along the rope of the two trefoil knots. Also shown are the topologically achiral figure eight knot "8", the simple link L, and the unknot U.

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