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When Alexander the Great cut the Gordian knot in the year 333 BCE, he almost certainly did not give a thought to the mathematical details of his (mis)deed. Similarly, when it comes to tying knots, neither scouts, mountaineers, fishermen, nor sailors care about the higher mathematics involved in this process. It was only because of an error that scientists were prompted to turn their attention to knots. This is what happened.

Toward the end of his career, the Scottish scientist Lord Kelvin (1824–1907) was of the belief that atoms consisted of fine tubes that would become tied up with each other and then buzz around the ether. Kelvin’s theory was generally accepted for about two decades before being proved erroneous. In the meantime, however, this mistaken belief had led Peter Tait (1831–1901), also a Scottish physicist, to categorize all possible knots. (Mathematical knots are different from their more mundane cousins in that both free ends are connected to each other. In other words, knots in knot theory are always closed loops.)

A superficial classification would use the number of crossings of two strands as determinants. This type of categorization does not account for the possibility, however, that two different-looking knots could actually be the same—that is, that one of them could be turned into the other one by picking, plucking, tugging, and pulling at, but without cutting or untying, their strands. Thus if one knot can be “deformed” into the other, the two are identical. Tait discovered this quite intuitively and attempted to account only for truly different knots in his scheme of classification. These so-called prime knots cannot be disassembled into further components.

Tait’s classification was not without its errors, though,

as Kenneth Perko, a New York lawyer, discovered in 1974. Working on his living room floor, he managed to turn one knot with 10 crossings into another knot that had been listed as different by Tait.

Nowadays we know that there exists only one knot with three crossings, another one with four crossings, and two with five crossings. Altogether there are 249 knots with up to 10 crossings. Beyond that the number of possibilities rises quickly. There are no less than 1,701,935 different knots with up to 16 crossings.

The central question in mathematical knot theory was and remains whether two knots are different or if one of them can be transformed into the other one without cutting and reattaching the strands. The transformation must be performed by means of three simple manipulations that were discovered by the German mathematician Kurt Reidemeister (1893–1971). A related question asks whether a tuft that looks like a knot is in fact an “unknot” because it can be disentangled by means of the Reidemeister manipulation. The well-known and well-worn trick of miraculously unknotting a complicated-looking tangle of strings that magicians use to their advantage under the astonished oohs and aahs of the audience is obviously based on an “unknot.”

Henceforth, mathematicians busied themselves looking for characteristic traits, so-called invariants, which could clearly and unambiguously be attributed to the various knots, thus making them distinguishable from one another. James Alexander (1888–1971), working at the IAS in Princeton, found polynomials (see footnote on page 28) that were suitable for the classification of knots. If the polynomials are different, the corresponding knots are also different. Unfortunately, it soon became apparent that the reverse does not hold true: Different knots may possess identical polynomials. Other mathematicians developed different systems of classification, and others still seek a workable recipe of how to convert identical knots from one form into another, equivalent, form.

Is this really relevant to anybody other than scouts, mountaineers, fishermen, or sailors? Knot theory is an

example of a mathematical subdiscipline that was developed before applications were even being considered. But with time, useful implementations of knot theory did surface and knots found applications in real life. Chemists and molecular biologists in particular became interested in knots. For example, some of them study the ways in which the long and stringy forms of the DNA molecule wind and twist so as to fit into the nucleus of a cell. If you were to enlarge a typical cell to the size of a football, the length of a DNA double helix would measure about 200 kilometers. And as everyone knows, long pieces of string have the annoying tendency of spontaneously becoming all twisted and tangled. What scientists are interested in is which forms of knots the DNA strings take on and how they then disentangle again.

And, of course, there are the theoretical physicists. Toward the end of last century it became evident that quantum mechanics and the force of gravity are not compatible. In the 1970s and 1980s quantum physicists suggested “string theory” as a new answer to this puzzle. This theory basically says that elementary particles are tiny little strings, crushed together in higher-dimensional spaces. (So maybe Lord Kelvin’s erroneous conjecture wasn’t so erroneous after all.) Obviously in this situation too, the strings get entangled, and so knot theory found another application.

There is yet another group of people—scientists included—who are interested in knot theory. It consists of the gentlemen who tie their neckties every morning. Thomas Fink and Yong Mao, two physicists at the Cavendish Laboratories in Cambridge, investigated the ways in which elegant men do up their ties before going to the office in the morning or to a dinner party in the evening. They found that no fewer than 85 different ways exist in which the task could be performed. Not all of them fulfill traditional aesthetic demands, however. There are, you see, a number of issues that need to be considered when performing what is commonly thought of as a rather routine act. For instance, the absolute sine qua non of the elegant knot is that it be symmetrical. Then, as all fashion-con-

scious men know, only the wider end of the tie may be moved around when tying the knot. And lastly, the number of times one moves the free end either to the right or to the left should be roughly equal. Hence, regrettably, trend-setting gentlemen who would like to adhere to these conditions are unable to take advantage of the full range of 85 possibilities. These unfortunate souls are left with merely 10 different knots to choose from.