Table of Knots

Knots are presented by their two dimensional projection indicating the over- and underpasses. Surely, every knot can be drawn in many different ways, with the different numbers of crossings, but the table uses for each knot the projection with minimal possible number of crossings. For instance, an unknot and a trefoil have minimal crossing numbers 0 and 3, respectively. 

Each knot in the table is denoted by its minimal crossing number. In most cases, there is more than one type of knot with any given minimal crossing number for instance, two knots with five crossings. These different 

Some knots are chiral, i.e., knot is different from its mirror image. For instance, there can be two distinct types of the trefoil, left and right, they cannot be transformed into one another continuously. Other knots, such as 4i, are not chiral, they are mirror symmetric. 

Equipped with this terminology, we can return to physics. 

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Rolfsen, D. Ainofs and Links Publish or Perish Berkeley, 1976 second printing with corrections Publish or Perish Houston, 1990, Appendix C Table of knots and links,... 

Statistical mechanics of knots. To solve the problem of statistical mechanics of knots, one needs, first of all, a knot invariant. Indeed a closed chain can be unknotted or can form knots of different types. The very beginning of the table of knots is shown in Fig.8. However, an analytical expression for the knot in variant is unknown. Therefore, we had to use a computer and analgebraic invariant elaborated in the topological theory of knots. We found that the most convenient in variant was the Alexander polynomial (reviewed by Frank-Kamenetsku and Vologodskh, 1981 [24] and Vologodskh and Frank-Kamenetskii, 1992 [81]). 

Returning to the story, W. Thomson got understandably excited and asked his friend and collaborator P.G. Tait (1831-1901) to make a table of possible knots and also try to compute the frequencies at which the knotted strings could oscillate — maybe, they hoped, it could explain the atomic spectra Tait worked hard, made first large table of knots and formulated several conjectures about classification of knots. The excitement among physicists continued for some years, but eventually nobody lesser than James Clerk Maxwell (1831-1879) grew sceptical, for there was no experimental support for the idea, and in 1878 he wrote in his letter to Tait ... 

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). 

Although we do not plan to delve into mathematics, a few words are necessary about classification of knots, just to introduce terms. The problem is that there is an incredible variety of different types of knots. Traditionally, starting from Tait, they are presented in the form of tables, similar to the one shown in the Figure Cl 1.2. 

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).

Table 12.1 Summary of knots that slipped completely and partially...

Table 12.3 Effect of knotting on tensile properties in dynamic tests...

The second item returned by the datafitQ function is a table containing three items a table of the x locations of the interpolation knots (the xc values), a table of the y values associated with the knots (the c values) and the uncertainties associated with the determination of the know y values from nlstsq() (the del values). In most used of this fimction this additional information will not be needed as the proof of the technique will simply reside in an observation of how well the returned function provides an acceptable smoofli curve representation of the input data points. 

Shorebirds use their sense of taste when probing sand for food. The purple sandpiper, Calidris maritima, and the knot, Calidris canutus, forage much longer in jars that contain food buried in sand, or sand with an extract of food, than in jars with plain sand (Gerritsen etal, 1983). Table 12.3 lists the responses of various seabirds to prey odors. 

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... 

Table 1. Kinetic, electrochemical, and photophysical properties of dicopper(I) knots containing polymethylene bridges.

Very different is the situation as far as the kinetic parameters of the complexes are concerned. The surprising inertness of the various dicopper(I) knots to de-metalation is clearly due to the knotted nature of their ligand. This is obvious from the data of Table 1. As far as (K-84)p is concerned, the amazing kinetic stability of its mono- and di-copper(I) complexes arises from topological and geome-... 

The nodules are found in gray-black commercial slate having fine grain size, uniform color and texture, and well developed slaty cleavage. Such slate occurs in thick strata marked only by thin black bands ( ribbons ) of somewhat coarser texture, and by rare, disseminated knots, siliceous nodules of foreign material. The slate is a mixture of quartz, illite, chlorite, caldte, and muscovite, with minor amounts of pyrite, carbonaceous matter, and heavy mineral grains. The dark color is attributed to finely disseminated carbon and pyrite. An analysis of the slate is given in Table I. 

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