The Tangled Task of Distinguishing Knots
Consider the plight of a gardener struggling with a recalcitrant tangle of garden hose. Sometimes, no amount of pulling or twisting unsnarls the coils. At other times, the tangles readily come apart, and the hose emerges unknotted.
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. 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.
Mathematicians can ask the same questions about a knotted curve that sailors or boy scouts may ask about a knotted rope. What kind of knot is it? Is the curve (or rope) really knotted? Can a second knot undo the first? Is one knot equivalent to another? How can different knots be distinguished?
To attack the problem of classifying or distinguishing knots, mathematicians have adopted a set of rules that make knots more convenient to study. Instead of analyzing three-dimensional knots, they examine two-dimensional shadows cast by these knots. Even the most tangled configuration can be shown as a continuous loop whose shadow winds across a flat surface, sometimes crossing over and sometimes crossing under itself. In drawings of mathematical knots, tiny breaks in the lines signify underpasses or overpasses.
One convenient measure of a knots complexity is the minimum number of crossings that show up after looking at all possible shadows of a particular knot. A loop without any twists or crossings (in its simplest form, a circle) is called an unknot.
The simplest possible true knot is the overhand, or trefoil, knot, which is really just a circle that winds through itself. In its plainest form, this knot has three crossings. It also comes in two forms: left-handed and right-handed configurations, which are mirror images of each other.
One approach to labeling knots is to use the arrangement of the crossings in a knot diagram to produce an algebraic expression for that knot. Such a label, which stays the same no matter how much a given knot may be deformed or twisted, is known as an invariant.
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To solve the problem of distinguishing among knots, mathematicians have tried to find schemes for labeling them in such a way that two knots having the same label are really equivalent–even when their diagrams may appear different–and that two knots with different labels are truly different. In the latter case, the label would be enough to indicate that no amount of twisting or tugging would ever transform one knot into the other.
Over the years, mathematicians have developed a variety of knot invariants, often expressed in the form of a polynomial algebraic expression.
In the 1920s, James W. Alexander (1888–1971) discovered a systematic procedure for generating such a formula from the pattern of over- and under-crossings in a knot diagram. Expressed as a positive or negative power of some variable with integer coefficients, his simple polynomials for characterizing and labeling knots turned out to be remarkably useful and relatively easy to compute.
If two knots have different Alexander polynomials, the knots are definitely not equivalent. For example, the trefoil knot carries the label t2 – t + 1 and the figure-eight knot is t2 – 3t + 1, and both differ from the unknot, whose polynomial is the constant 1.
But the method isnt foolproof. Knots that have the same Alexander polynomial arent necessarily equivalent. The procedure doesnt distinguish, for example, between the granny knot and the square knot even though it is impossible to deform one into the other.
For a long time, the Alexander polynomial was one of the few tools topologists had for telling knots apart. In 1984, however, knot theorists were suddenly and unexpectedly thrust into new mathematical territory overrun with novel invariants.
The mathematician who triggered the stampede was Vaughan F.R. Jones of the University of California, Berkeley. He found a completely new invariant: a new polynomial that does a better job than the Alexander polynomial at distinguishing knots.
In the new scheme, the Jones polynomial for the unknot is 1. It is t + t3 – t4 for the trefoil knot. A systematic procedure (algorithm) allows the Jones polynomial to be computed for any knot, based on its pattern of crossings.
Joness discovery prompted a great deal of excitement in the mathematics community because his polynomial detects the difference between a knot and its mirror image, something that the Alexander polynomial had failed to do. The finding was also surprising because Jones unexpectedly discovered a connection between von Neumann algebras (mathematical techniques that play a role in quantum mechanics) and braid theory.
A braid can be thought of as a set of hanging strings that have been interlaced in some pattern. The top and bottom ends of such a pattern can be connected together to form a knotted mathematical braid.
The Jones polynomial apparently encodes many kinds of knot characteristics. Indeed, mathematicians are amazed that a single polynomial can incorporate so much knot information. However, computing the Jones polynomial for messy tangles can be enormously difficult. Indeed, researchers have proved that as knots get more and more complicated, the difficulty of computing their Jones polynomials increases exponentially.
Mathematicians strongly suspect that the Jones invariant and its many siblings are part of a still bigger picture that they barely yet glimpse. Because the current invariants still cant distinguish certain classes of distinct knots, they also know that the picture is far from complete.
Like physicists who are trying to make sense of the particles and forces that make up the physical world, knot theorists are looking for something akin to a grand unified theory that would explain all invariants and all knots. They hope eventually to find the ultimate invariant–one that truly distinguishes any two distinct knots.