Mathematicians are studying knottiness:
Untying a math mystery, by Margaret Wertheim, LA Times: ...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 ... 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 ... 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, ... Millett helped to revitalize knot theory when he was part of a team that discovered ... 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. ...