Tying up a knotty loose end.

Tying up a knotty loose end

One way to picture a mathematical knot is to think of a tangled string with its two ends spliced together. IN more abstract terms, it can be thought of as a one-dimensional curve twisting through three-dimensional space to form a closed loop. Knottedness, however, is not a property of the curve itself. An imaginary ant crawling along a narrow tunnel within the one-dimensional confines of such a curve, even after completing its circuit, would never be able to tell whether the curve is knotted. Instetad, knottedness resides in the way the curve sits in three-dimensional space. That relationship between a knot and the space in which it sits has long intrigued mathematicians and has suggested several important questions that play a central role in knot theory.

One problem, first considered in 1908 by topologist Heinrich Tietze, concerns determining whether two knots that look different are really the same knot when untangled. One knot could be merely a twisted, stretched version of the other. Tietze conjectured that if the space around two knots is the same, then the two knots themselves are tied in teh same way. Eighty years later, Cameron M. Gordon of the University of Texas at Austin and John E. Luecke of New York University's Courant Institute finally proved Tietze right: No essential information about a knot is lost by throwing away the knot and, instead, studying and manipulating the space around it.

The theorem justifies a commonly used strategy for distinguishing knots. Whereas a knot can be thought of as a twisted loop of string, the space around a knot -- the knot's complement -- can be pictured as a slab of jelly from which a thin tube has been extracted. Instead of working with knots themselves, mathematicians often manipulate and deform the corresponding complements to derive mathematical expressions useful for characterizing and sometimes distinguishing knots. Matehmaticians already could prove that two knots are not the same simply by proving that their complements are not the same. Now they know that if two complements are the same, the corresponding knots are also the same.

The idea that knots with identical complements would also be the same sounds obvious. But, in the case of two or more knots linked together, mathematicians have already proved that links having identical complements are not necessarily equivalent. That curious behavior provided one clue that the problem of single knots and their complements was mathematically quite subtle.

Indeed, Gordon and Luecke's proof takes up most of a 59-page manuscript. "It's a simply stated problem, but it turned out to be tricky to prove," Gordon says. "One approaches the problem in a somewhat roundabout way, and you end up proving something that seems to be a little bit stronger." Their proof involves the consequences of a mathematical process known as Dehn surgery, in which a knot is fattened into a tube, removed from its complement and then stitched back in again.

To arrive at the proof, Gordon and Luecke developed techniques that reduced the problem to studying sets of labeled diagrams. In recent years, that kind of approach, in which problems are reformulated in terms of questions about appropriate diagrams, has turned out to be useful for a variety of mathematical proofs. "We would hope that what we have done already, with a little bit more work, will give some information on other problems," Gordon says. "We're trying to refine the techniques now."

Gordon and Luecke would like to use such techniques to explore the mysterious behavior or linked knots. "It would be nice to explore where these arguments lead, but that's still rather speculative," Gordon says. "You never know in mathematics until you really get down to it whether a method will work for a given problem."