Loopy Solution Brings Infinite Relief

Circle game. In time, three basic maneuvers will reduce a tangle, like the one at left, to a simple loop.

Twenty-three centuries after Alexander the Great slashed through the Gordian knot, mathematicians have finally made their first stab at figuring out how long it takes to untie one key class of knots. The unheroic answer is "not forever"--and even that comes with a huge string attached. Still, knot researchers are delighted.

As far back as the 1920s, mathematicians had figured out that untangling a knot and putting it in a standard, recognizable form requires only three types of motion--the so-called Reidemeister moves. But there is no easy way to tell how many Reidemeister moves it takes to untangle any loop of string and render it recognizable--even one as simple as mere loop of string, called an unknot, that has been tangled up a bit to disguise it.

Now two knot theorists have unravelled the problem. Jeffrey Lagarias of AT&T Labs in Florham Park, New Jersey, and Joel Hass of the University of California, Davis, considered an unknot as the boundary of a crumpled and distorted disk, rather than as a twisted-up loop of string. They performed Reidemeister-equivalent operations on the disk and translated it back into knot form. Their conclusion: A finite number of Reidemeister moves will untangle any given twisted-up unknot, they report in the current issue of the Journal of the American Mathematical Society. Not that the solution is especially practical--if the string in a knot crosses itself n times, they guarantee that you can untangle it in 2(100,000,000,000n) Reidemeister moves. In other words, if every atom in the universe were doing a googol googol googol Reidemeister moves a second from the beginning of the universe to the end of the universe, that wouldn't even approach the number you need to guarantee unknotting a singly twisted rubberband.

"The [bound] is, of course, enormous and hopeless," Lagarias admits. But he says that just putting a cap on it may inspire future researchers to whittle it down to a reasonable size. Indeed, the question of whether a limit even existed at all was "a very big problem," says Joan Birman, a knot theorist at Barnard College in New York City.

Related sites
Lagarias's home page
An introduction to knot theory

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