Excerpt from Five More Golden Rules : Knots, Codes, Chaos and Other Great Theories of 20Th-Century Mathematics
By Casti, John L.
Chapter 1: The Alexander Polynomial
Knot History, Mathematics
According to legend, one of the world's first knot theorists was the simple Greek peasant, Gordius, who was named king of Phrygia when he arrived in a public square in an oxen cart around 1200 B.C., thus fulfilling an oracle's pronouncement that the future king would arrive in a wagon. In founding the Anatolian city of Gordium, the peasant gave thanks to the gods by dedicating his oxcart to Zeus, and tying it with an unusual knot. Then, as now, oracles were in ample supply, and another soothsayer came forward to claim that whoever untied the knot would rule all of Asia (at that time, meaning Persia). History records that in 333 B.C., Alexander the Great reached Gordium, was shown Gordius's chariot, and immediately "untied" the intricate knot with one fell swoop of his sword, thus giving rise to the proverbial term "solving a problem by bold action." This ancient story also gives rise to the principal problem facing knot theorists today: How difficult is it to untie a knot?
As with a lot of modern mathematics, the first important study of knots is due to Gauss, who while investigating electromagnetism in the latter part of the nineteenth century, developed methods for calculating properties of knots that he identified with the linking of electric and magnetic fields. A bit later, Lord Kelvin proposed the notion that atoms could be classified according to different types of knots, which led mathematicians such as Tait and Kirkman to produce tables of knot types. These efforts underscore the crucial problem of determining when one knot is the same as another. In what follows, we shall focus mostly on this question, as it remains the most important question in knot theory and is still only partially solved.
Amusing as it is for historical reasons, the Gordian knot is pretty far from the thoughts of theorists today interested in developing methods for distinguishing one knot from another. Knot theorists are motivated by far more important-or at least intellectually more challenging-questions than who is to rule Persia. For example, the strands of DNA residing at the center of every cell of your body are tightly coiled and tangled when packed into genes. But to replicate itself in the process of cell division, a DNA strand must uncoil itself. This unknotting process is carried out by enzymes that serve as microscopic versions of Alexander's sword, slicing through the strands of DNA to unknot them, and then "gluing" the ends back together again. So being able to say how difficult it is to untie a knot can help in determining how fast these enzymes can carry out their activity-as well as help us develop entirely new enzymes that might do things better, faster, or both. Much the same story can be told about questions in polymer chemistry. Even in the rarefied heights of elementary particle physics, there is growing evidence that the constituents of matter itself are nothing more than ultramicroscopic strings of energy coiled up in particular configurations. We'll see these and other examples of ways that knot theory works in practice later in the chapter. But for now let's take a harder look at what mathematicians mean when they talk about a "knotty" problem.
Have Knot, Will Unravel
Intuitively, a knot K is simply a closed curve sitting in ordinary three-dimensional space. Unfortunately, this definition is not quite accept-able because it admits offbeat curves containing an infinite sequence of smaller and smaller knottings, as well as other undesirable pathologies. So it's technically better to define a knot as a simple, closed curve composed of a finite-but possibly very large-number of straight line segments. It can be shown that all the properties of continuous curves we want for studying knots can be recovered from this definition by let-ting the number of segments approach infinity. So when pictures in the chapter show knots as continuous curves, bear in mind the straight-line definition underlying the figures.
A lowly cockroach crawling along any knot will see it as being just a one-dimensional closed curve and will have no sense of the three-dimensional space in which the knot sits (technically, its embedding). So it is the way the curve sits in an embedding space that distinguishes one knot from another. But why must the curve be embedded in three-dimensional space, rather than the plane, or even in four-dimensional space? The answer is that the plane is too small to contain a knot because the places where the curve crosses over itself to form the knot would have to stick up out of the plane. On the other hand, four-dimensional space is too big because any knot can be unraveled there, given the extra degree of freedom. So just as with Goldilock's porridge, ordinary three-dimensional space is not too big and not too small, but just right as the embedding space within which to talk about interesting knots. The simplest possible knot, which is not really a knot at all, but the opposite of a knot-the unknot-is shown in part (a) of Figure 1.1. Part (b) shows the simplest nontrivial knot, the trefoil or overhand knot. And just to show how complicated things can get, part (c) of the figure shows a nonstandard picture of the same trefoil knot. This illustrates nicely why there is a problem in trying to distinguish one knot from another, because it is far from obvious that the two pictures of the trefoil really represent exactly the same knot.....
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