Topologically achiral

We shall now see how to apply the theorem to the molecular trefoil knot, which was illustrated in Figure 17. We can create a molecular cell complex G by replacing each isolated benzene ring by a cell and each chain of three fused rings by a single cell. We prove by contradiction that our molecular cell complex is topologically chiral. Suppose that it is topologically achiral. Then there is a defor-... 

Figure 27. A topologically achiral embedding of a Mobius ladder with four rungs.

An object, such as a knot, or a link, or a graph, is topologically chiral if and only if it cannot be converted into its mirror image by continuous deformation (ambient isotopy) in the object s space otherwise it is topologically achiral. 

The knots in Figure 21 are all prime knots because they cannot be divided (factored) into smaller, nontrivial knots. Prime knots are the building blocks of composite knots and of links. Like prime numbers, which yield composite numbers upon multiplication, or like atoms in chemistry, which yield molecules upon combination, prime knots are the elementary units of knot theory. Composite knots are exemplified by the topologically achiral square knot and the topologically chiral granny knot (Figure 22). In each of these knots, a plane perpendicular to the... 

Figure 23. Rigidly achiral presentations of some topologically achiral prime knots, (a) 4i (S4 symmetry) (b) 63 (S4 symmetry) (c) 12427 (Si = Ci symmetry) 12toi9 ( 6 symmetry) (e) 818 (Si symmetry) (f) IO123 (Sin symmetry).

With few exceptions, all nonoriented chemical links reported in the literature are topologically achiral. One of the rare exceptions is the two-component 4-crossing link 4 (Figure 31).114 The molecule in the depicted conformation, and indeed in any imaginable conformation, is geometrically chiral. It is therefore reasonable to conjecture that it is also topologically chiral. Proof that 4 is indeed topologically chiral was provided after the development of suitable polynomials.115... 

In support of his proposal, Tauber pointed out that For certain knots Ze = 0. This is exactly as it should be, for precisely these knots are identical with their mirror images. Similarly, Walba asserted that The number of 8s and >cs are summed arithmetically. If there are the same number of 8 and X crossings, then the knot must be topologically achiral. Contrary to these assertions, however, alternating knots whose writhe is zero are not necessarily amphicheiral The simplest example is knot 84. Nineteen of the 32 10-crossing prime knots with writhe zero are topologically chi ral, and 13 of these are alternating.144 Two hundred... 

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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