Composite trefoil knot

Fig. 3 Elements of molecular templates (a) a crossover reacts with a linker to give (i) a catenane and (ii) a rotaxane (b) a fused turn closed with a fused linker gives a macrobicycle (c) two anchors and two pairs of inverted ditopic turns template a trefoil knot (d) composite knots require two two-anchor templates and (e) oligocatenanes can be formed with multicrossover templates on a single anchor.

Pig. 11.1 Several simple knots. For the trefoil knot 3i, two isomers are shown, they are mirror impages of one another. Also, two distinct composite knots are shown (called granny and square), combining different trefoil isomers. 

Table of knots includes only prime knots. Obviously, several knots can be tied on a single rope, in which case one talks of a composite knot (this is very similar to prime and composite numbers). Figure 11.1 shows two ways to combine two trefoils. Importantly, one can easily convince oneself that there is not such thing as an anti-knot or knot annihilation given any knot on the rope, one cannot tie another knot on the same rope such as to make the composite knot an unknot (in this sense composition of knots is like multiplication of integers, there is no analog of division, therefore, no inverse). 

Figure 4-28. Two representations of a trefoil knot (A and B) and two related composite knots (C and D).

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