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 one-dimensional 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 much-longed-for "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 eighth-graders 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.