Magicians are experts at tying knots that look intractable yet unravel on command. Befuddled spectators find it difficult to distinguish between such phony tangles and truly knotted ropes.
Mathematicians also tussle with knots, but their task has an additional constraint. Unlike a knotted piece of rope, a mathematical knot has no free ends.
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In this context, a knot is a one-dimensional curve that winds through itself in three-dimensional space, finally catching its tail to form a closed loop. You can untie a shoelace and untangle a fishing line, but you can’t get rid of the knot in a mathematician’s loop without cutting the strand.
If a particular tangled loop doesn’t really have a knot in it and the loop can be unraveled and smoothed out to a circle, mathematicians call the configuration an unknot.
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Determining at a glance or two whether a given tangled loop is a knot or an unknot can be as difficult for mathematicians as it is for spectators of a masterly magician’s knotty prestidigitations.
Knot theorists have long sought practical procedures for distinguishing knotted curves from unknotted ones. Two recent developments provide some new hints.
This research activity is just one thread of a resurgent interest in mathematical knots, not only for mathematicians but also among other scientists. Molecular biologists have used insights from knot theory to understand how DNA strands can be broken and then recombined into knotted forms (SN: 11/16/96, p. 310). Other investigators have explored potential roles for knotted loops in theoretical physics (SN: 5/3/97, p. 270).
Knot or unknot?
One way to tell whether a certain tangled loop is really an unknot is to model it out of string, then try twisting and pulling it in various ways. If you manage to untangle the loop, you know it’s an unknot.
Failure to untangle the loop even after hours of fruitless labor, however, doesn’t prove that the loop is truly knotted. It’s possible you somehow overlooked the right combination of manipulations to undo the tangle.
To distinguish among complicated loops, mathematicians imagine knots to be constructed out of perfectly flexible, stretchable, and infinitesimally thin string. For convenience, they focus on the shadow cast by such loops on a flat, two-dimensional surface. These shadows, technically called projections, are often drawn with small breaks indicating where one part of the loop crosses over or under another part.
Knot theorists use the number of crossings in such a diagram as one way to characterize a given knot projection. Depending on the viewpoint and configuration, the same knot or unknot can be represented by many different projections, which may also have different numbers of crossings.
It’s easy to tell that a knot projection with just two crossings is always an unknot. The problem of determining what a given diagram represents becomes increasingly difficult as the number of crossings goes up.
In 1926, mathematician Kurt Reidemeister proved that if you have two distinct projections of the same knot, you can get from one projection to the other using a sequence of basic moves. There are three such operations, and they are now known as Reidemeister moves. In the case of an unknot, some combination of these fundamental moves will inevitably untangle even a messy loop. Finding the appropriate moves for unwinding a complicated configuration, however, is generally no simple matter.
That it is always possible to identify an unknot, even without specifying the precise sequence of Reidemeister moves, was firmly established in 1961 by Wolfgang Haken of the University of Illinois at Urbana-Champaign. He came up with a procedure for deciding whether a given knot is really an unknot.
Haken’s algorithm involves not the tangled loop itself but the imagined surface for which the loop serves as a boundary. To visualize such a surface, consider the soap film that spans a ring or a twisted loop after it emerges from a soap solution. In the case of a ring, which represents the circular form of the unknot, the surface is simply a flat disk. Twisted loops that are not actually knotted also serve as boundaries of disklike surfaces—but here the disks may be extremely convoluted.
Haken’s method mathematically replicates the process of flattening out an exceedingly crumpled surface to end up with a flat disk. If the process succeeds, the original tangled loop was certainly an unknot.
The method that Haken formulated, however, is so complicated that no one has yet been able to write a fully practical computer program to follow the necessary steps and, for a given projection, give a “yes” or “no” answer to the question of whether it’s an unknot.
Several groups have tried to implement Haken’s algorithm. These implementations “are too bulky at present to allow for analysis of more than a very small number of crossings,” says mathematician Joel Hass of the University of California, Davis.
In a striking new development, Haken’s approach has proved useful in tackling another important question: What are the maximum number of Reidemeister steps that would remove all the crossings in a given projection of a tangle to arrive at an unknot?
In general, there’s no easy way to tell in advance how many moves it would take to untangle a given loop and render it recognizable. Moreover, it isn’t obvious that the number of required moves is always finite.
Now, Hass and Jeffrey C. Lagarias of AT&T Labs-Research in Florham Park, N.J., have for the first time established a ceiling on the number of Reidemeister moves required to unravel a tangle. The mathematicians reported their findings in the February Journal of the American Mathematical Society.
Like Haken, Hass and Lagarias looked at the mathematical characteristics of convoluted disklike surfaces to obtain their proof. They established that if a string crosses itself n times, you can untangle it in no more than 2(100,000,000,000n) Reidemeister moves.
Although this ceiling is an enormous number, just putting a cap on the untangling process represents a notable achievement, comments knot theorist Colin C. Adams of Williams College in Williamstown, Mass.
“The bound is currently so large that no algorithm using these ideas will be forthcoming for quite a while, but perhaps the upper bound can be brought down,” he notes. “The hardest step, getting any bound at all, has been done.”
Mathematicians have already started to lower the ceiling. Topologist Andrew R. Casson of Yale University, for instance, has proposed one possible avenue that would significantly reduce the upper bound on the number of moves.
While mathematicians now know an upper bound on the number of Reidemeister moves, they still do not know which ones to use and where to use them. In other words, says Joan S. Birman of Barnard College at Columbia University, “they do not have an algorithm based upon Reidemeister moves.”
Mathematicians have also sought more tractable alternatives to the Haken algorithm for recognizing an unknot. One promising scheme was originally proposed in 1998 by Birman and computer scientist Michael D. Hirsch, then at Emory University in Atlanta.
Birman and Hirsch took advantage of the fact that knots can be represented as braids. You can picture a mathematical braid as a set of strings, each one attached to a bar at the top and another bar at the bottom. Each string heads downward but may cross over or weave among the other strings along the way before reaching the bottom. You can then pull the bottom bar around and glue it to the top bar, connecting the strings in corresponding locations on the bars to form a closed braid. It turns out that every such braid corresponds to a knot, an unknot, or a set of linked knots.
In general, mathematicians have found that braids are easier to understand and work with than knots. Moreover, they have a convenient scheme for coding a braid diagram in terms of a few numerical parameters, which enables them to readily reconstruct a knot from those values.
Departing significantly from Haken’s approach, Birman and Hirsch’s novel algorithm for identifying unknots is built upon the idea of considering a knot or an unknot as a closed braid. In the Birman-Hirsch scheme, the braid’s parameters can indicate whether it represents an unknot. A closed braid with given parameters serves as the boundary of a wrinkled disk with a particular type of folding and layering. Such layering patterns simplify the mathematicians’ task.
Initially, it wasn’t entirely clear whether this idea could be translated into a workable recipe for identifying an unknot. Birman and her collaborators Marta Rampichini, Paolo Boldi, and Sebastiano Vigna of the University of Milan in Italy have now worked out a way to perform the systematic enumerations of unknots required to make the scheme practical.
“It’s definitely exciting to have an approach other than Haken’s,” Adams remarks. However, “we have yet to see whether this will be a fruitful approach computationally,” he adds.
Foolproof identification of unknots is only the first step, however. “The unknot is simply the easiest knot to study,” Birman says.
“Ultimately,” Adams notes, “we would like an algorithm and computer program that would tell us whether any two knots are equivalent.” That goal, however, remains elusive.
Mathematicians have already developed some very fast, practical algorithms for distinguishing knots that belong to certain large classes, but these methods do not cover every possible type of knot.
Identification and classification of knots is linked to an intriguing notion that geometrical structures known as 3-manifolds could serve as models of physical space in the universe (SN: 2/21/98, p. 123: http://www.sciencenews.org/sn_arc98/2_21_98/bob1.htm).
The general theory of relativity posits that gravity is essentially a geometric effect—in other words, the theory links mass with the local curvature of space. Interestingly, it says nothing about the shape of the universe—the overall form, or topology, of the three-dimensional spatial component of relativity’s four-dimensional space-time.
An important example of a 3-manifold is the so-called complement of a knot, which can be imagined as all of space minus the knot. It’s almost as if a microscopically thin worm had bored a hole through space leaving a twisty tunnel in its wake.
“Knot complements are the most easily visualized 3-manifolds, so their study is a fundamental aspect of 3-manifold topology,” Birman says.
In the course of developing computer programs to help survey all types of 3-manifolds, mathematicians have inevitably also delved into the business of distinguishing knots. For example, the program known as SnapPea, developed by freelance geometer Jeffrey R. Weeks of Canton, N.Y., has been a remarkably successful tool for identifying knots. Although there is no mathematical guarantee that the method works in every case, it has yet to fail for an important class of knots, Hass notes.
As a result of such mathematical efforts, cosmologists have all sorts of candidates for the topology of the universe. It’s conceivable, for example, that the cosmos has a weirdly convoluted structure modeled on some sort of knot complement.
Studies of braids themselves have proved helpful in elucidating phenomena such as intriguing twists in the rings of Saturn. Knots arise in the study of equations describing weather systems and in other mathematical models used in physics.
“Often the knots play a fairly subtle role,” Birman says.
With intriguing applications in molecular biology and theoretical physics, along with surprising new results in solving long-standing mathematical conundrums, knot theory promises to keep mathematicians tied up for many years to come.