Trefoil left handed

Unknot Right-handed trefoil Left-handed trefoil... 

Figure 13. Diagrams of left-handed (L.) and right-handed (R) trefoil knots.

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.

Figure 5.16 A direction-independent SGIM characterization of a space curve C, regarded as a molecular backbone. On the left-hand side the shape globe S of radius R is shown enclosing the space curve C. The centre of the sphere is chosen as the centre of mass of chain molecule C. On the right-hand side the shape invariance domains of the sphere are shown, as defined by the knot types derived from the projections. There are only two knot types in this example unknots and trefoil knots.

There are two chiralities of trefoil knots right-handed and left-handed, depending on overcrossing handedness along the oriented curve (see Figure 6). These two knots are not distinguished by the polynomial A(t). For complicated knots, the computation of the entire polynomial is feasible, but it can be a formidable task. i... 

The symbols in (6) are a short-hand notation for the knots found [20d]. The notations 0i,3i,3i, and 4j identify the unknot, left-handed trefoil knot, right-handed trefoil knot, and the figure-eight knot, respectively. The corresponding Jones polynomials are as follows [14] ... 

If the three di-metal complexes of Figure 11 display classical chirality because of the presence of a double helix as core of the system, this is no longer true for the products obtained after demetallation. The trefoil knot is the only chiral demetallation pr uct, its rotation being determined by the right- or left-handed nature of its double helix precursor, as shown in Figure 12. 

Topological chirality [8,45] implies that there is no other way for the molecule to racemize than breaking a chemical bond. Knots represent beautiful examples of such systems. In addition to their fascinating topological properties related to chirality, they may also serve as interesting chiral systems for enantioselective interactions and processes (electron transfer interaction with other molecules, in particular, DNA incorporation of knots into the structure of molecular catalysts etc...). Consequently, and as a first step, it is particularly challenging to separate the left- and right-handed forms of our synthetic trefoil knot. 

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