Knot Theory

knot theory, in mathematics, the study of closed curves in three dimensions, and their possible deformations without one part cutting through another. Knots may be regarded as formed by interlacing and looping a piece of string in any fashion and then joining the ends. The first question that arises is whether such a curve is truly knotted or can simply be untangled; that is, whether or not one can deform it in space into a standard unknotted curve like a circle. The second question is whether, more generally, any two given curves represent different knots or are really the same knot in the sense that one can be continuously deformed into the other.

The basic tool for classifying knots consists of projecting each knot onto a plane—picture the shadow of the knot under a light—and counting the number of times the projection crosses itself, noting at each crossing which direction goes “over” and which goes “under.” A measure of the knot’s complexity is the least number of crossings that occur as the knot is moved around in all possible ways. The simplest possible true knot is the trefoil knot, or overhand knot, which has three such crossings; the order of this knot is therefore denoted as three. Even this simple knot has two configurations that cannot be deformed into each other, although they are mirror images. There are no knots with fewer crossings, and all others have at least four.

The number of distinguishable knots increases rapidly as the order increases. For example, there are almost 10,000 distinct knots with 13 crossings, and over a million with 16 crossings—the highest known by the end of the 20th century. Certain higher-order knots can be resolved into combinations, called products, of lower-order knots; for example, the square knot and the granny knot (sixth-order knots) are products of two trefoils that are of the same or opposite chirality, or handedness. Knots that cannot be so resolved are called prime.

The first steps toward a mathematical theory of knots were taken about 1800 by the German mathematician Carl Friedrich Gauss. The origins of modern knot theory, however, stem from a suggestion by the Scottish mathematician-physicist William Thomson (Lord Kelvin) in 1869 that atoms might consist of knotted vortex tubes of the ether, with different elements corresponding to different knots. In response, a contemporary, the Scottish mathematician-physicist Peter Guthrie Tait, made the first systematic attempt to classify knots. Although Kelvin’s theory was eventually rejected along with ether, knot theory continued to develop as a purely mathematical theory for about 100 years. Then a major breakthrough by the New Zealand mathematician Vaughan Jones in 1984, with the introduction of the Jones polynomials as new knot invariants, led the American mathematical physicist Edward Witten to discover a connection between knot theory and quantum field theory. (Both men were awarded Fields Medals in 1990 for their work.) In another direction, the American mathematician (and fellow Fields medalist) William Thurston made an important link between knot theory and hyperbolic geometry, with possible ramifications in cosmology. Other applications of knot theory have been made in biology, chemistry, and mathematical physics.