Typing knots to tubular geometry.

Sailors and other knot users have a large repertoire from which to choose the right knot for any purpose, whether it's tying a shoelace, mooring a boat, or knotting together bedsheets to escape from an upper story of a building. Mathematicians also tangle with diverse knots, but they have traditionally concerned themselves with those tied in a one-dimensional string with no free ends, forming closed loops.

Like sailors and scouts, mathematicians are interested in identifying and classifying different knots (SN: 5/21/88, p. 328). Now, a team of researchers has proposed a novel method of characterizing knots, based on imagining them as tubes pulled tight into the configurations that give the highest possible ratio of volume to surface area. These tubular objects, in turn, serve as useful mathematical models of knotted DNA loops and polymer chains.

Structural biologist Andrzej Stasiak of the University of Lausanne, Switzerland, and his colleagues report their findings in the Nov. 14 Nature.

"Such methods potentially offer a practical approach to [knot] classification," says mathematician Jonathan K. Simon of the University of Iowa in Iowa City. The methods may also "lead us to understand and predict how different knot types behave in real physical situations."

To tackle the problem of identifying and distinguishing knots, mathematicians have generally looked at the minimum number of times a given loop, laid on a flat surface, crosses over or under itself. The simplest possible knot is the trefoil knot, which is really just a loop that winds through itself once. In its plainest form, this knot has three crossings. Only one knot has four crossings, and two distinct knots have five crossings.

Over the years, mathematicians have worked out several different ways to use arrangements of crossings to generate the algebraic formulas that serve as labels for knots. Such a label, which stays the same no matter how a given knot may be deformed or twisted, is known as an invariant (SN: 10/26/85, p. 266). However, because two different knots may occasionally have the same formula, this scheme isn't foolproof.

Stasiak and his group approached the problem from a new perspective. They tied their knots not from imaginary one-dimensional strings, which have no width, but from tubes of a uniform diameter. They defined the ideal knot of a given type as that made from the shortest piece of tube.

The researchers used computer simulations to test their idea, starting with a narrow, knotted tube, then uniformly expanding the tube's diameter until there was contact between different portions of the tube.

"The modeled knots usually 'flowed' into their ideal geometrical representations, repeatedly adopting almost identical configurations [for a given knot]," the researchers report. Their method handled knots containing as many as 10 crossings.

The computer simulations revealed that these ideal knots have interesting mathematical properties. For example, the length-to-diameter ratio of a tubular knot is directly related to the average number of crossings, independent of a knot's actual shape.

Stasiak and his coworkers also demonstrated in their simulations that the average shape of loosely knotted DNA loops flopping around in a good solvent closely resembles the ideal configuration of the given knot.