Knots are everywhere. Sailors use them to sail the seas; Girl and Boy Scouts learn them to earn merit badges.
They're also an increasingly important branch of mathematics - knot theory - that could enrich sciences ranging from subatomic physics to chemistry to heredity.Five years after Vaughan Jones of the University of California-Berkeley revolutionized the mathematical study of knots, his work is helping physicists develop the Holy Grail of physics, a "theory of everything" - superstring theory - that might explain all known forces and particles.
That word comes from no less a figure than the guru of superstring theory, Ed Witten of Princeton, whose advocacy has helped make it one of the trendiest ideas in university physics departments.
"This knot theory work has opened a mysterious new world that has shed new light on some of the puzzles of string theory . . . an amazing new light on some of the mathematics of strings," Witten said. "To me, it's the most exciting thing going on in string theory."
Jones' work on knots could also help chemists to design unusual new molecules and geneticists to unravel secrets of life.
Jones, 36, a New Zealand native, arrived at Berkeley in 1984 and "immediately astounded us and the mathematical world with his discoveries about knots," said Irving Kaplansky, director of the Mathematical Sciences Research Institute in Berkeley.
In January, Witten and many other experts gathered at the university to discuss Jones' knot research.
The excitement was "feverish," Kaplansky said. "Eighteen people assembled and knocked each others' brains out."
Knot theory is a branch of a mathematical subject known as topology.
Topologists are interested in the properties of shapes that remain the same after they have been twisted or contorted.
For example, imagine an object made of rubber and shaped like a teacup. With enough squeezing and twisting, the object could be reshaped into a doughnut. The one property that remains common to the two objects is their hole - the hole in the doughnut and the hole in the teacup's handle.
Likewise, knot theorists are interested in features that remain the same as a knot is reshaped.
Consider two hypothetical knots, which we'll call A and B. At first, A and B may appear totally different; one might look quite simple while the other resembles a spaghetti-like snarl. But with enough manipulation, the thicker knot may be untangled until it resembles the other. If so, then the two knots are the "same" in a topological sense.
Is there a good way to tell whether two knots are topologically similar? Indeed there is - with so-called mathematical objects called polynomials.
In 1928, the mathematician John Alexander found there is a polynomial associated with each knot; these are now called Alexander polynomials. Also, similar knots may have similar polynomials. An improved version of the Alexander polynomial was developed in 1963 by the mathematician John Conway.
But the turning point came in 1984, when Jones' discovery "triggered the stampede," as Ivars Peterson writes in his book "The Mathematical Tourist" (W.H. Freeman, 1988).
"Knot theorists were suddenly and unexpectedly thrust into new mathematical territory . . . "
While working on an unrelated area of math called von Neumann algebras, Jones developed a new polynomial that, among other things, allows one to distinguish - mathematically speaking - between a knot and its mirror image.
"Very interesting," a layperson may yawn, "but so what?"
Here's what (among other things): The latest and potentially most far-reaching development in knot theory is its impact on the much-publicized concept of superstrings.
For decades, physicists have hoped to unify all known forces and subatomic particles in a single grand equation, jokingly called the "theory of everything."
In recent years, one of the top candidates for the theory of everything has been superstring theory.
According to this theory:
- Nature consists ultimately of infinitesimally tiny, wiggling, string-shaped bundles of energy called superstrings.
- The universe consists of numerous different subatomic particles, like quarks, electrons, and neutrinos - each the result of a superstring that vibrates in a particular manner.
In the theory, "depending on how the string vibrates, one gets a different particle - just as one gets a different note depending on how one plucks a violin string," says a superstring theorist, Orlando Alvarez of UC-Berkeley.
Superstring theory is exciting because "it's the only theory we know that seems to unify gravity and quantum mechanics in a consistent way."
Now, Alvarez says, knot research "has shown it may be possible to formulate these (superstring) theories in a completely different way . . . a new way (for physicists) to approach relations between superstring theory and what are called `completely integral' models - that is, models (of nature) that have a large number of conservation laws, like the conservation of electric charge or the conservation of energy," Alvarez said.
Researchers began speculating about links between knot theory and superstrings about two years ago, Alvarez said.
Excitement heated up last summer when Witten made some important findings in the area.
"Jones' work shows there are interconnections between areas that people would never have thought there were interconnections - and Witten's (subsequent) work (on knots and superstrings at Princeton) shows there are even more interconnections," Alvarez said.
Still, Alvarez cautioned, "we have to wait a little while before we can assess the full impact" of knot theory on superstring theory.
Here's one problem: to verify superstring theory would require an accelerator (a.k.a. "atom smashers") or as-yet unknown device capable of accelerating subatomic particles to energies that are, for now, unimaginably high.
Meanwhile, chemists are also benefiting from Jones' knot theory, according to Peterson's book. They hope to develop knot-shaped chains of molecules never before seen. Such molecules would be more flexible than the stiff molecules assembled according to old-fashioned "Euclidean" geometry.