For about a century, the mathematical theory of knots has been dealing with “embedding the unit circle into three-dimensional space.” A knot is mathematically defined as “a closed, piecewise linear curve in three-dimensional Euclidean space.” Mathematical knot theory is a branch of topology that focuses on idealized strings, which are assumed to be infinitely thin. Knot theory not only interests mathematicians but, for once, also fascinates laypeople since it is easy to visualize threads and strings as real objects. The fact that knot theory relates to three dimensions is yet another point in its favor. If one takes knot theory into the realms of four-dimensional space, then all knots—which have been tied using one-dimensional strings—instantly become “unknots.”

The physical theory of knots, in contrast to the mathematical version, deals not with infinitely thin abstractions but rather with real ropes that possess a finite diameter or thickness. Scientists dealing with the physical theory of knots are interested, for example, in what sorts of knots can be tied in the real world. Or they investigate how much rope is needed to tie a specific knot. Current thinking is that the length needed to tie a specific knot might be a measure of its complexity. Since stringlike objects, such as DNA, possess finite size, the physical theory of knots can provide much more realistic answers than the abstract mathematical theory to scientific problems.

When dealing with real knots, the actual configuration of the strings is of the utmost importance. In mathematical knot theory, all knots that can be transformed into each other by means of tugging, twisting, and pulling are considered identical. This is not true for the physical theory. Here the exact positioning of the strings is of

crucial importance. Any deviation in the arrangement of the pieces of rope, however minute, produces a new knot. In other words, whenever one tugs at a knot, a new knot appears. Each knot has an infinite number of appearances. This difficulty is the reason that seemingly simple problems remain unsolved to this very day.

Take the simplest knot of all, the trefoil or overhand knot. Until very recently nobody knew whether a rope measuring 1 inch in diameter and 1 foot in length could be tied into a trefoil knot. (In knot theory the two ends of the rope must be attached to each other; that is, the rope forms a closed loop. Thus the trefoil knot becomes a cloverleaf knot.)

Simple reflection reveals that a rope which is only π (approximately equal to 3.14) times longer than it is thick does not suffice to tie any knot. One can form no more than a compact ring by linking together the two loose ends. (The length is measured in the middle of the rope.) There is no rope left over for tying the actual knot! Thus π is a lower bound for any knot. Knowing this fact does not answer the question, however, of what the minimum length is to tie a string into a cloverleaf knot. (This is, incidentally, where the builders’ cocky retort must derive when asked how long a job will take. “Well, how long is a piece of string?” is the answer, which means, in other words, “Who knows?”)

To achieve some progress, knot theorists took recourse to a bright idea. They designed a computer model that depicts knots, with the assumption that repelling energy is distributed along the rope. As a consequence, the strands would be pushed away from each other and the knot would transform itself into a configuration in which the strands are separated from each other as far as possible. Any slack in the strands would become visible and could then be eliminated by pulling and tugging. Based on these and similar exercises, mathematicians continued to seek the minimal length required to tie a knot.

In 1999 four scientists succeeded in calculating a new lower bound for the rope length. They established that even if the rope measures about 7.8 times its thickness

(2.5 times π), it is not sufficiently long to tie a cloverleaf knot. A few years later three other researchers were able to prove that a ratio of length over thickness of 10.7 does not suffice either. It was only in 2003 that Yuanan Diao, a Chinese scientist then at the University of North Carolina, was able to come up with the answer to the original question. It was negative: Diao proved that even 12 inches of a 1-inch-thick rope does not suffice to tie a cloverleaf knot. At the same time he worked out a formula that calculated lower bounds of rope lengths for all knots with up to 1,850 crossings.

Later Diao managed to refine the conditions for a cloverleaf knot even further. He demonstrated that at least 14.5 inches of rope was necessary to tie that particular knot. On the other hand, computer simulations showed that 16.3 inches would suffice to do the job. Obviously, the truth lies somewhere in between these two figures.

Another question with which knot theorists of the physical persuasion grapple refers to the mysterious and complex form of the legendary Gordian knot. Alexander the Great could only undo it with the stroke of his sword. What was its exact configuration? For a long time it had been assumed that the knot was tied when the rope was still wet and that it had then been left to dry in the sun. The knotted rope would have then shrunk to its minimum length. In 2002 the Polish physicist Piotr Pieranski and the biologist Andrzej Stasiak from the University of Lausanne, Switzerland, found such a knot. With the help of computer simulations they were able to create a knot whose rope length was too short to untie. In their statement to the press they said that “the shrunken loop of rope was entangled in such a way that it could not be converted back to its original circle by simple manipulations.”

Working on these computer simulations, the two researchers made yet another and wholly unexpected discovery that may turn out to have far-reaching consequences. They defined a “winding number” for knots: Each time one strand of the rope is strung over the other strand from left to right, the number 1 is added. Whenever the

strand is strung over the other strand from right to left, the number 1 is subtracted. To their immense surprise, the average winding number—where the average is taken over all viewing angles—amounted to a multiple of the fraction 4/7 for each and every knot they subjected to the calculations. No explanation has so far been offered for this phenomenon. Recall in this context that “string theory” describes elementary particles as small, possibly entangled strings. Hence, some physicists suspect that the quantified properties of elementary particles may rest on this rather mysterious particularity of “knot quanta.”

Physical knots find very real applications in daily life, for example, in tying shoelaces. Burkhard Polster, a mathematician at Monash University in Australia, decided to subject this mundane routine to rigorous mathematical analysis. The criteria he used were the length of the lace, the firmness of the binding, and the tightness of the knot. On the assumption that each eyelet of the shoe contributes to the tension of the lacing, Polster proved that the least amount of lace is required when the laces are crossed not every time but every other time. (The precise number of crossings is a function of whether the number of eyelets is an even or odd number.)

It stands to reason that this way of tying up one’s shoes does not really provide a lot of firmness. If the tension at the back of the foot is of importance, then the traditional ways of tying the shoes are certainly the best: Either you cross the laces each time they are thread through the eyelets or—another traditional and possibly more elegant way of doing up your shoes—you thread one end of the lace from the bottom eyelet into the top eyelet on the opposite side and thread the other end in parallel strips from one side to the other.

Once the shoe has been laced, what is the recommended way to tie the loose ends into a knot? Most people make a double knot, and the loops serve merely as decoration. But here too things are not as straightforward as one might assume initially. There are, it turns out, two ways of tying this knot, and the difference between them could not be more obvious. One knot is the granny knot, where

both ends of the lace are crossed over twice in the same direction. Every boy and girl scout knows that this knot is not sufficiently tight. Proof of this is provided on each and every playground, with mothers forever having to bend down to do it all over again. (No wonder Velcro is so popular. Unfortunately, it deprives children of one of the most exciting learning experiences.) A much tighter and more durable knot is the so-called square knot. It is very similar to the granny knot, with one crucial difference. The knot is first tied by crossing the laces in one direction and then, moving to the second knot, by crossing the two loops in the opposite direction.