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of the mathematical entities known as von Neumann algebras. Unexpectedly, Jones noticed that a relation that surfaced in von Neumann algebras looked suspiciously similar to a relation in knot theory, and he met with Columbia University knot theorist Joan Birman to discuss possible applications. An examination of that relation eventually revealed an entirely new invariant for knots, dubbed the Jones polynomial. The Jones polynomial was immediately recognized as a more sensitive invariant than the Alexander polynomial. It distinguishes, for instance, between knots and their mirror images (e.g., the right-handed and left-handed trefoil knots in figure 57), for which the Alexander polynomials were identical. More importantly, however, Jones’s discovery generated an unprecedented excitement among knot theorists. The announcement of a new invariant triggered such a flurry of activity that the world of knots suddenly resembled the stock exchange floor on a day on which the Federal Reserve unexpectedly lowers interest rates.

Figure 57

There was much more to Jones’s discovery than just progress in knot theory. The Jones polynomial suddenly connected a bewildering variety of areas in mathematics and physics, ranging from statistical mechanics (used, for instance, to study the behavior of large collections of atoms or molecules) to quantum groups (a branch of mathematics related to the physics of the subatomic world). Mathematicians all over the world immersed themselves feverishly in attempts to look for even more general invariants that would somehow encompass both the Alexander and Jones polynomials. This mathematical race ended up in what is perhaps the most astonishing result in the history of scientific competition. Only a few months after Jones revealed his new polynomial, four groups, working independently and using three different mathematical approaches, announced at the same time the discovery of an even more sensitive invariant. The new polynomial became known as the HOMFLY polynomial, after the first letters in the names of the discoverers: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. Furthermore, as if four groups crossing the finish line in a dead heat weren’t enough, two Polish mathematicians (Przytycki and Traczyk) discovered independently precisely the same polynomial, but they missed the publication date due to a capricious mail system. Consequently, the polynomial is also referred to as the HOMFLYPT (or sometimes THOMFLYP) polynomial, adding the first letters in the names of the Polish discoverers.

Since then, while other knot invariants have been discovered, a complete classification of knots remains elusive. The question of precisely which knot can be twisted and turned to produce another knot without the use of scissors is still unanswered. The most advanced invariant discovered to date is the work of the Russian-French mathematician Maxim Kontsevich, who received the prestigious Fields Medal in 1998 and the Crafoord Prize in 2008 for his work. Incidentally, in 1998, Jim Hoste of Pitzer College in Claremont, California, and Jeffrey Weeks of Canton, New York, tabulated all the knotted loops having sixteen or fewer crossings. An identical tabulation was produced independently by Morwen Thistlethwaite of the University of Tennessee in Knoxville. Each list contains precisely 1,701,936 different knots!

The real surprise, however, came not so much from the progress in knot theory itself, but from the dramatic and unexpected comeback that knot theory has made in a wide range of sciences.

The Knots of Life

Recall that knot theory was motivated by a wrong model of the atom. Once that model died, however, mathematicians were not discouraged. On the contrary, they embarked with great enthusiasm on the long and difficult journey of trying to understand knots in their own right. Imagine then their delight when knot theory suddenly turned out to be the key to understanding fundamental processes involving the molecules of life. Do you need any better example of the “passive” role of pure mathematics in explaining