Is God a Mathematician?

by Mario Livio

  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 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 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.

  Since strings are closed loops moving through space, as time progresses, they sweep areas (known as world sheets) in the form of cylinders (as in figure 60). If a string emits other strings, this cylinder forks to form wishbone-shaped structures. When many strings interact, they form an intricate network of fused donutlike shells. While studying these types of complex topological structures, string theorists Hirosi Ooguri and Cumrun Vafa discovered a surprising connection between the number of donut shells, the intrinsic geometric properties of knots, and the Jones polynomial. Even earlier, Ed Witten—one of the key players in string theory—created an unexpected relation between the Jones polynomial and the very foundation of string theory (known as quantum field theory). Witten’s model was later rethought from a purely mathematical perspective by the mathematician Michael Atiyah. So string theory and knot theory live in perfect symbiosis. On one hand, string theory has benefited from results in knot theory; on the other, string theory has actually led to new insights in knot theory.

  Figure 60

  With a much broader scope, string theory searches for explanations for the most basic constituents of matter, much in the same way that Thomson originally searched for a theory of atoms. Thomson (mistakenly) thought that knots could provide the answer. By a surprising twist, string theorists find that knots can indeed provide at least some answers.

  The story of knot theory demonstrates beautifully the unexpected powers of mathematics. As I have mentioned earlier, even the “active” side of the effectiveness of mathematics alone—when scientists generate the mathematics they need to describe observable science—presents some baffling surprises when it comes to accuracy. Let me describe briefly one topic in physics in which both the active and the passive aspects played a role, but which is particularly remarkable because of the obtained accuracy.

  A Weighty Accuracy

  Newton took the laws of falling bodies discovered by Galileo and other Italian experimentalists, combined them with the laws of planetary motion determined by Kepler, and used this unified scheme to put forth a universal, mathematical law of gravitation. Along the way, Newton had to formulate an entirely new branch of mathematics—calculus—that allowed him to capture concisely and coherently all the properties of his proposed laws of motion and gravitation. The accuracy to which Newton himself could verify his law of gravity, given the experimental and observational results of his day, was no better than about 4 percent. Yet the law proved to be accurate beyond all reasonable expectations. By the 1950s the experimental accuracy was better than one ten-thousandth of a percent. But this is not all. A few recent, speculative theories, aimed at explaining the fact that the expansion of our universe seems to be speeding up, suggested that gravity may change its behavior on very small distance scales. Recall that Newton’s law states that the gravitational attraction decreases as the inverse square of the distance. That is, if you double the distance between two masses, the gravitational force each mass feels becomes four times weaker. The new scenarios predicted deviations from this behavior at distances smaller than one millimeter (the twenty-fifth part of an inch). Eric Adelberger, Daniel Kapner, and their collaborators at the University of Washington, Seattle, conducted a series of ingenious experiments to test this predicted change in the dependence on the separation. Their most recent results, published in January 2007, show that the inverse-square law holds down to a distance of fifty-six thousandths of a millimeter! So a mathematical law that was proposed more than three hundred years ago on the basis of very scanty observations not only turned out to be phenomenally accurate, but also proved to hold in a range that couldn’t even be probed until very recently.

  There was one major question that Newton left completely unanswered: How does gravity really work? How does the Earth, a quarter million miles away from the Moon, affect the Moon’s motion? Newton was aware of this deficiency in his theory, and he openly admitted it in the Principia:

  Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the Sun and planets…and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances…But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses.

  The person who decided to meet the challenge posed by Newton’s omission was Albert Einstein (1879–1955). In 1907 in particular, Einstein had a very strong reason to be interested in gravity—his new theory of special relativity appeared to be in direct conflict with Newton’s law of gravitation.

  Newton believed that gravity’s action was instantaneous. He assumed that it took no time at all for planets to feel the Sun’s gravitational force, or for an apple to feel the Earth’s attraction. On the other hand, the central pillar of Einstein’s special relativity was the statement that no object, energy, or information could travel faster than the speed of light. So how could gravity work instantaneously? As the following example will show, the consequences of this contradiction could be disastrous to concepts as fundamental as our perception of cause
and effect.

  Imagine that the Sun were to somehow suddenly disappear. Robbed of the force holding it to its orbit, the Earth would (according to Newton) immediately start moving along a straight line (apart from small deviations due to the gravity of the other planets). However, the Sun would actually disappear from view to the Earth’s inhabitants only about eight minutes later, since this is the time it takes light to traverse the distance from the Sun to the Earth. In other words, the change in the Earth’s motion would precede the Sun’s disappearance.

  To remove this conflict, and at the same time to tackle Newton’s unanswered question, Einstein engaged almost obsessively in a search for a new theory of gravity. This was a formidable task. Any new theory had not only to preserve all the remarkable successes of Newton’s theory, but also to explain how gravity works, and to do so in a way that is compatible with special relativity. After a number of false starts and long wanderings down blind alleys, Einstein finally reached his goal in 1915. His theory of general relativity is still regarded by many as one of the most beautiful theories ever formulated.

  At the heart of Einstein’s groundbreaking insight lay the idea that gravity is nothing but warps in the fabric of space and time. According to Einstein, just as golf balls are guided by the warps and curves across an undulating green, planets follow curved paths in the warped space representing the Sun’s gravity. In other words, in the absence of matter or other forms of energy, spacetime (the unified fabric of the three dimensions of space and one of time) would be flat. Matter and energy warp spacetime just as a heavy bowling ball causes a trampoline to sag. Planets follow the most direct paths in this curved geometry, which is a manifestation of gravity. By solving the “how it works” problem for gravity, Einstein also provided the framework for addressing the question of how fast it propagates. The latter question boiled down to determining how fast warps in spacetime could travel. This was a bit like calculating the speed of ripples in a pond. Einstein was able to show that in general relativity gravity traveled precisely at the speed of light, which eliminated the discrepancy that existed between Newton’s theory and special relativity. If the Sun were to disappear, the change in the Earth’s orbit would occur eight minutes later, coinciding with our observing the disappearance.