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By Root 703 0

line, in terms of his role in the development of knot theory, was the German “Prince of Mathematics,” Carl Friedrich Gauss. Several of Gauss’s notes contain drawings and detailed descriptions of knots, along with some analytic examinations of their properties. As important as the works of Vandermonde, Gauss, and a few other nineteenth century mathematicians were, however, the main driving force behind the modern mathematical knot theory came from an unexpected source—an attempt to explain the structure of matter. The idea originated in the mind of the famous English physicist William Thomson, better known today as Lord Kelvin (1824–1907). Thomson’s efforts concentrated on formulating a theory of atoms, the basic building blocks of matter. According to his truly imaginative conjecture, atoms were really knotted tubes of ether—that mysterious substance that was supposed to permeate all space. The variety of chemical elements could, in the context of this model, be accounted for by the rich diversity of knots.

If Thomson’s speculation sounds almost crazy today, it is only because we have had an entire century to get used to and test experimentally the correct model of the atom, in which electrons orbit the atomic nucleus. But this was England of the 1860s, and Thomson was deeply impressed with the stability of complex smoke rings and their ability to vibrate—two properties considered essential for modeling atoms at the time. In order to develop the knot equivalent of a periodic table of the elements, Thomson had to be able to classify knots—find out which different knots are possible—and it was this need for knot tabulation that sparked a serious interest in the mathematics of knots.

As I explained already in chapter 1, a mathematical knot looks like a familiar knot in a string, only with the string’s ends spliced. In other words, a mathematical knot is portrayed by a closed curve with no loose ends. A few examples are presented in figure 54, where the three-dimensional knots are represented by their projections, or shadows, in the plane. The position in space of any two strands that cross each other is indicated in the figure by interrupting the line that depicts the lower strand. The simplest knot—the one called the unknot—is just a closed circular curve (as in figure 54a). The trefoil knot (shown in figure 54b) has three crossings of the strands, and the figure eight knot (figure 54c) has four crossings. In Thomson’s theory, these three knots could, in principle, be models of three atoms of increasing complexity, such as the hydrogen, carbon, and oxygen atoms, respectively. Still, a complete knot classification was badly needed, and the person who set out to sort the knots was Thomson’s friend the Scottish mathematical physicist Peter Guthrie Tait (1831–1901).

The types of questions mathematicians ask about knots are really not very different from those one might ask about an ordinary knotted string or a tangled ball of yarn. Is it really knotted? Is one knot equivalent to another? What the latter question means is simply: Can one knot be deformed into the shape of the other without breaking the strands or pushing one strand through the other like a magician’s linking rings? The importance of this question is demonstrated in figure 55, which shows that by certain manipulations one can obtain two very different representations of what is actually the same knot. Ultimately, knot theory searches for some precise way of proving that certain knots (such as the trefoil knot and the figure eight knot; figures 54b and 54c) are really different, while ignoring the superficial differences of other knots, such as the two knots in figure 55.

Figure 54

Tait started his classification work the hard way. Without any rigorous mathematical principle to guide him, he compiled lists of curves with one crossing, two crossings, three crossings, and so on. In collaboration with the Reverend Thomas Penyngton Kirkman (1806–95), who was also an amateur mathematician, he started sifting through the curves to eliminate duplications by equivalent