Untying a math mystery
By Margaret Wertheim
MARGARET WERTHEIM is the science columnist for the
LA Weekly and director of the Institute for Figuring, which has recently
been hosting a series of lectures on knot theory.
March 6, 2006
YOU HAVE to hand it to mathematicians, they can turn anything into a formal
problem. Balls packed into boxes, folded paper, even bits of string become,
in the hands of mathematical theorists, gateways to worlds of Byzantine
complexity and beauty.
Take a piece of string — I mean literally, go get a piece of string
and tie it into a knot. Now tape the two ends together so it makes a closed
loop — necessary to fulfill the mathematical definition of a "knot."
How many different knot types do you think there are? The number is infinite,
and the question of how to categorize these manifestations of loopiness
has engaged some of the finest mathematical minds for a century.
We are nowhere near to having a complete taxonomy of knots, and some mathematicians
view the problem as so inherently difficult that they think it is an impossible
goal. Indeed, "knot theory" is an area of mathematics in which
almost any generalized question you can think of is unlikely to be answered.
Although knots in math are essentially onedimensional objects, understanding
them has turned out to be a significant challenge.
Moreover, knots provide mysterious links between the mathematical continents
of topology, geometry and algebra, hinting that these enigmatic twists
contain secrets to powerful, deep and general truths.
And yet this most esoteric branch of mathematics has also turned out to
have immense application in the physical world. That's because we now
know that DNA and many other long molecules arrange themselves into knotted
structures. Knot theorists are suddenly in demand from biologists, who
want help understanding how clumps of DNA move through different mediums,
how proteins fold up and how polymers behave. The specific knottiness
of a piece of DNA, for example, determines whether certain enzymes can
act on it, which has important implications for understanding diseases
such as cancer.
Ken Millett, a knot theorist at UC Santa Barbara, is a leader in the application
of this mathematics to DNA and other molecules. In the 1980s, inspired
by UC Berkeley mathematician Vaughan Jones, Millett helped to revitalize
knot theory when he was part of a team that discovered a strange new way
of classifying knots. With this method, each knot can be associated with
a particular equation that uniquely characterizes it. Still, mathematicians
have no idea what the equations actually signify; they don't seem to relate
to any of the usual features of knots, such as shape and form. "Do
they refer to some hidden structure within the knot?" Millett asks.
"We really don't know."
Some physicists, however, think the equations are telling us something
fundamental about the basic particles and forces of nature. They believe
these arcane formulas may enable us to find the muchlongedfor "theory
of everything" under the umbrella of string theory. The equations
also turn out to have application to the emerging field of quantum computing,
which many scientists hope will usher in an era of new, more powerful
computational devices.
The story of knots suggests that we never know from what areas of mathematics
useful applications may spring. Although mathematics has no physical substance,
it can be as precious as gold or oil, and ultimately as integral to our
economy. As President Bush noted in January's State of the Union speech,
America's place at the top of the global technological pyramid depends
on a workforce that is well educated in math and science. Yet, nationally,
our schools are understaffed in these critical areas. Which brings me
to the importance of Millett's other professional hat — math education.
In addition to his knot research, Millett directs a program at Santa Barbara
that recruits math and science undergraduates to become classroom teachers.
Given that a recent report by the National Academy of Sciences revealed
that nearly 60% of American eighthgraders are taught math by teachers
who did not major in math or pass any kind of certification exam, efforts
such as Millett's are critical. On Feb. 25, his work was honored in Washington
with an award from the organization Quality Education for Minorities.
In the State of the Union address, the president pledged to train 70,000
math and science teachers to handle AP courses. But the plan does not
call for hiring any new teachers, which is woefully shortsighted. Math
education does not require expensive equipment, specialized buildings
or fancy facilities, it just needs good teachers and a supportive learning
environment.
The lessons of knot theory suggest that investing in this "arcane"
subject will, in the long run, pay dividends.

