Knotty views.

Mathematical research resembles the construction of an intricate framework in a fogenshrouded environment. Guided by the basic rules of logic, workers use a variety of tools to assemble components of a great but largely obscured structure. Occasionally the fog lifts just enough to reveal unsuspected links between disparate elements.

A decade ago, few mathematicians would have predicted that the study of knots could furnish a unifying thread in mathematical research. But the fog has gradually thinned, revealing a surprisingly extensive web connecting knot theory with various mathematical specialties.

"Knots are turning up all over in mathematics," says Joan S. Birman, a mathematician at Columbia University in New York City.

Moreover, new developments in knot theory have provided valuable insights into various aspects of physics, chemistry and biology. In particular, researchers have identified deep connections between the problem of characterizing knots and several areas of mathematics and physics that no one previously suspected had any place for knots (SN: 5/21/88, p.328).

Birman and colleague Xiao-Song Lin of Columbia have added one more strand to this knotted web by bringing new mathematical techniques to bear on the study of knots. Birman described these developments at a joint meeting of the American Mathematical Society and the Mathematical Association of America, held in January in Baltimore.

Compared with the antiquity of many basic ideas in mathematics, knot theory is relatively young. The initial impetus for the systematic study of knots came from a suggestion made more than a century ago concerning the structure of matter. At that time, physicist William Thomson, who later took on the title Lord Kelvin, imagined atoms as minute, doughnut-shaped vortexes of swirling fluid embedded in a pervasive, spazce-filling medium called the ether.

To explain what distinguishes one chemical element from another, Thomson turned to the notion of a knot. He envisioned atoms of different elements as distinctively knotted vortex tubes. Each twisted tube looked like a knotted rope with its two ends joined together in a loop to keep the knot from coming apart.

Intrigued by this idea, Thomson's colleague Peter G. Tait set out to discover what kinds of knots were possible. This monumental, trial-and-error effort resulted in the first tables of knots, organized according to the minimum number of crossings evident in diagrams of the two-dimensional shadows cast by three-dimensional knotted loops.

However, because the same knot can be pictured in two dimensions in many different ways, this undertaking foundered on the difficulty of determining whether the lists were really complete. The researchers had no foolproof method of testing wheter two knots, as represented by their diagrams, were the same or actually wound through space in fundamentally different ways.

To solve the problem of distinguishing among knots, mathematicians tried to develop schemes for labeling them in such a way that two knots having the same label are really equivalent -- even though their diagrams may appear quite different -- and two knots with different labels are truly different.

One such method involves using the crossings in a knot diagram to derive a numerical or algebraic expression that serves as label for the knot. Such a label, which stays the same no matter how much a given knot may be deformed or twisted, is known as a knot invariant.

In 1984, Vaughan F. R. Jones of the University of California, Berkeley, unexpectedly discovered a connection between knot theory and mathematical techniques that play a role in quantum mechanics. This discovery led to the formulation of a host of new algebraic invariants (or knot polynomials), computed from knot diagrams, that distinguish among knots more effectively than earlier schemes (SN: 10/26/85, p.266), which sometimes gave the same label to knots known on other grounds to be different.

Although mathematicians had recipes for computing these new invariants, they had little sense of what features of three-dimensional knots the resulting algebraic expressions encoded. Even the subsequent discovery of a link between these knot invariants and quantum field theory, which tries to account for interactions between elementary particles, proved unenlightening to many mathematicians (SN: 3/18/89, p.174).

Two years ago, Victor A. Vassiliev of the Russian Academy of Sciences in Moscow introduced a new, radically different way of looking at knots. He started by considering a huge, multidimensional, mathematical "space," in which each point represents a possible three-dimensional knot configuration. If two knots are equivalent to each other, there exists a pathway in this abstract space from one configuration to the other.

This strategy allowed Vassiliev to study not just individual knots but also the ways in which distinct knots fit together. Indeed, his attempt to classify pathways from one knot to its equivalent led to a means of computing numerical knot invariants associated with patterns of connected lines -- known as graphs -- in which crossed strands in a knot diagram merge to form nodes.

Initially, Vassiliev's approach seemed formidable and impractical. Many mathematicians who read his paper found his techniques very difficult to apply in practice and could see no guarantee that usable knot invariants would emerge from his work.

Birman and Lin, however, discovered a way of translating Vassiliev's scheme into a set of rules and a list of potential starting points. "That's what began to suggest that [Vassiliev's invariants] really looked like the knot invariants we already knew," Birman says.

News of this work brought Dror Bar-Natan, now at Harvard University but then a student at Princeton, into the picture. After several days of discussions with Bar-Natan, Birman and Lin proved that the Jones invariants and several related expressionsare are directly connected with Vassiliev's knot labels. Bar-Natan discovered simultaneously a remarkable link between his own work on Feynman diagrams - pictures used to provide an intuitive interpretation of interactions between subatomic particles - and Vassiliev's original equations for computing invariants.

Although this research doesn't completely solve the problem of how to interpret the numerous knot invariants that mathematicians have discovered, it provides a familiar framework within which they can begin to tackle the problem. "It changes an old problem you didn't know how to do into a new, hard problem that's a lot of work," Birman says. "It's a beginning."

"Vassiliev's work provides a very good insight into the nature of knot invariants generally," says Louis H. Kauffman of the University of Illinois at Chicago. "It's entirely possible that all of the invariants we know are built of the building blocks coming out of Vassiliev's picture."

"It gives us another unifying principle for describing knot polynomials," Birman adds. "Instead of one explanation for knot polynomials, we are instead finding multiple explanations and interrelationships, each very beautiful and each opening new doors for investigation."