Prime knot

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... 

In priming plastic explosives with primacord, the triple roll knot (see Figure 5) is a good one to use. Three turns of the cord are concentrated in one tight knot, insuring detonation of the charge. 

The charge may be primed with a blasting cap or a primacord triple roll knot. The cap or knot should be embedded in the center of the explosive behind the plate but not pushed in so far that it touches the plate. See Figure 37. 

Many types of mortar shells are packed unassembled in separate containers. In such cases one need only fill the fuse well with plastic explosive and prime it with a primacord triple roll knot. Others, including U.S. shells, are packed fully assembled. Following are the steps to be taken to prepare a fully assembled shell for use as an ambush charge ... 

If the booster is not used, the warhead may be primed by filling the cavity in the top with plastic explosive, into which is embedded and fastened a blasting cap or a primacord knot. 

Prime the charge either with a triple roll prima-cord knot or a nonelectric blasting cap. Figures 64 and 65 illustrate properly made up charges. 

C.C. Adams, The Knot Book. W.H. Freeman and Co., New York, 1994, p. 33. In addition to the prime knots listed we expect several dozen composite knots for n= 12. 

Figure 2. The first twenty-five prime knots.

The second equation poses no problem because 8 was defined as a phase function. If T) < 1, the solution for S is trivial, as the standard field with form. st then becomes a knot. The same happens if q is bounded, say, if q <4, because we can then take as the Clebsch variables q = q/ , 8 = n 8, where n is an integer greater than A. Dropping the primes and entering the new Clebsch variables in (144), it is clear that there then exists a solution for S, y. 

Figure 10.85 The first four prime knots. The number denotes the number of crossings, while the subscript is the order of the knot. (Reprinted with permission from [98]).

Figure 21. Diagrams of prime knots with up to seven crossings.

The existence of a rigidly achiral presentation suffices as proof of the knot s amphicheirality because all chiral presentations can be isotoped to their mirror images by way of the achiral one. The only possible point group for rigidly achiral presentations of prime (but not composite) knots13 is S2 , n- 1,2,. Figure 23 depicts diagrams of such presentations for a selected number of amphicheiral prime... 

Census of Chiral and Achiral Prime Knots and Alkanes"... 

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).

Like knots, links may be prime or composite. The Hopf and Borromean links are examples of prime links because they cannot be divided (factored) into smaller, nontrivial links. Figure 30(a) is the minimal diagram of a composite link that is the abstract representative of some [3]catenanes, one of which is depicted in Figure 30(b).103b That the three-component link is a composite link is shown by the fact that a plane perpendicular to the plane of projection (dashed line) and pierced in exactly two points cuts the link in half If the open ends on both sides of the plane are now joined to form closed curves, two Hopf links result. In analogy to composite knots, the three-component link in Figure 30(a) is denoted by 2 2j, and the five-component composite link that represents olympiadane by 2 2 2 2. ... 

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... 

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). 

Fig. C11.2 This table shows all possible prime knots with up to 8 crossings on the projection. For chiral knots, only one of the mirror images is shown. There are many ways to identify certain separate classes of knots for instance, knots 3i, 5i, 7i are called torus knots, because they can be nicely placed on the surface of a doughnut (and there is obviously a torus knot with any odd number of crossings). But the classification of knots relevant for their physics is yet to be developed. The figure is courtesy of R. Scharein the knot images were produced by his software KnotPlot (see http //www.knotplot.com).

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