Is God a Mathematician? (28 page)

A breakthrough in knot theory came in 1928 when the American mathematician James Waddell Alexander (1888–1971) discovered an important invariant that has become known as the
Alexander polynomial.
Basically, the Alexander polynomial is an algebraic expression that uses the arrangement of crossings to label the knot. The good news was that if two knots had different Alexander polynomials, then
the knots were definitely different. The bad news was that two knots that had the same polynomial could still be different knots. While extremely helpful, therefore, the Alexander polynomial was still not perfect for distinguishing knots.

Mathematicians spent the next four decades exploring the conceptual basis for the Alexander polynomial and gaining further insights into the properties of knots. Why were they getting so deeply into that subject? Certainly not for any practical application. Thomson’s atomic model had long been forgotten, and there was no other problem in sight in the sciences, economics, architecture, or any other discipline that appeared to require a theory of knots. Mathematicians were spending endless hours on knots simply because they were curious! To these individuals, the idea of understanding knots and the principles that govern them was exquisitely beautiful. The sudden flash of insight afforded by the Alexander polynomial was as irresistible to mathematicians as the challenge of climbing Mount Everest was to George Mallory, who famously replied “Because it is there” to the question of why he wanted to climb the mountain.

In the late 1960s, the prolific English-American mathematician John Horton Conway discovered a procedure for “unknotting” knots gradually, thereby revealing the underlying relationship between knots and their Alexander polynomials. In particular, Conway introduced two simple “surgical” operations that could serve as the basis for defining a knot invariant. Conway’s operations, dubbed
flip
and
smoothing,
are described schematically in figure 56. In the flip (figure 56a), the crossing is transformed by running the upper strand under the lower one (the figure also indicates how one might achieve this transformation in a real knot in a string). Note that the flip obviously changes the nature of the knot. For instance, you can easily convince yourself that the trefoil knot in figure 54b would become the unknot (figure 54a) following a flip. Conway’s smoothing operation eliminates the crossing altogether (figure 56b), by reattaching the strands the “wrong” way. Even with the new understanding gained from Conway’s work, mathematicians remained convinced for almost two more decades that no other knot invariants (of the type of the Alexander polynomial) could be found. This situation changed dramatically in 1984.

Figure 56

The New Zealander–American mathematician Vaughan Jones was not studying knots at all. Rather, he was exploring an even more abstract world—one 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 pre
cisely 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 nature?

Deoxyribonucleic acid, or DNA, is the genetic material of all cells. It consists of two very long strands that are intertwined and twisted around each other millions of times to form a double helix. Along the two backbones, which can be thought of as the sides of a ladder, sugar and phosphate molecules alternate. The “rungs” of the ladder consist of pairs of bases connected by hydrogen bonds in a prescribed fashion (adenine bonds only with thymine, and cytosine only with guanine; figure 58). When a cell divides, the first step is replication of DNA, so that daughter cells can receive copies. Similarly, in the process of
transcription
(in which genetic information from DNA is copied to RNA), a section of the DNA double helix is uncoiled and only one DNA
strand serves as a template. After the synthesis of RNA is complete, the DNA recoils into its helix. Neither the replication nor the transcription process is easy, however, because DNA is so tightly knotted and coiled (in order to compact the information storage) that unless some unpacking takes place, these vital life processes could not proceed smoothly. In addition, for the replication process to reach completion, offspring DNA molecules must be unknotted, and the parent DNA must eventually be restored to its original configuration.

Figure 58

The agents that take care of the unknotting and disentanglement are enzymes. Enzymes can pass one DNA strand through another
by creating temporary breaks and reconnecting the ends differently. Does this process sound familiar? These are precisely the surgical operations introduced by Conway for the unraveling of mathematical knots (represented in figure 56). In other words, from a topological standpoint, DNA is a complex knot that has to be unknotted by enzymes to allow for replication or transcription to occur. By using knot theory to calculate how difficult it is to unknot the DNA, researchers can study the properties of the enzymes that do the unknotting. Better yet, using experimental visualization techniques such as electron microscopy and gel electrophoresis, scientists can actually observe and quantify the changes in the knotting and linking of DNA caused by an enzyme (figure 59 shows an electron micrograph of a DNA knot). The challenge to mathematicians is then to deduce the mechanisms by which the enzymes operate from the observed changes in the topology of the DNA. As a byproduct, the changes in the number of crossings in the DNA knot give biologists a measure of the
reaction rates
of the enzymes—how many crossings per minute can an enzyme of a given concentration affect.

But molecular biology is not the only arena in which knot theory
found unforeseen applications. String theory—the current attempt to formulate a unified theory that explains all the forces in nature—is also concerned with knots.

Figure 59

The Universe on a String?

Gravity is the force that operates on the largest scales. It holds the stars in the galaxies together, and it influences the expansion of the universe. Einstein’s general relativity is a remarkable theory of gravity. Deep within the atomic nucleus, other forces and a different theory reign supreme. The strong nuclear force holds particles called
quarks
together to form the familiar protons and neutrons, the basic constituents of matter. The behavior of the particles and the forces in the subatomic world is governed by the laws of quantum mechanics. Do quarks and galaxies play by the same rules? Physicists believe they should, even though they don’t yet quite know why. For decades, physicists have been searching for a “theory of everything”—a comprehensive description of the laws of nature. In particular, they want to bridge the gap between the large and the small with a quantum theory of gravity—a reconciliation of general relativity with quantum mechanics. String theory appears to be the current best bet for such a theory of everything. Originally developed and discarded as a theory for the nuclear force itself, string theory was revived from obscurity in 1974 by physicists John Schwarz and Joel Scherk. The basic idea of string theory is quite simple. The theory proposes that elementary subatomic particles, such as electrons and quarks, are not pointlike entities with no structure. Rather, the elementary particles represent different modes of vibration of the same basic string. The cosmos, according to these ideas, is filled with tiny, flexible, rubber band–like loops. Just as a violin string can be plucked to produce different harmonies, different vibrations of these looping strings correspond to distinct matter particles. In other words, the world is something like a symphony.