July 28, 1996|By Frank D. Roylance | Frank D. Roylance,SUN STAFF

The school bus is on its way, and your kid comes to you with a knot in his shoelace. It's a doozy. Fat and chaotic. Gordian

maybe, demanding a solution with the edge of a swift, sharp blade.

Draw your sword if you must, but don't mention it to John Conway. Knots "are my friends," he says.

John H. Conway, 58, is the John von Neuman professor of mathematics at Princeton University. He is also a knot theorist, knot evangelist, knot showman and master of endless knot parlor tricks. He can tie impossible knots without letting go of the string, and challenges rubes to make knots that just leave them in a tangle.

"Magic tricks are all part of teaching," he says. He uses them often in his lectures on knot theory. "I feel when you're teaching somebody, you should be enthusiastic. And if you can't be enthusiastic, you can fake it."

Last February, Conway captivated a room full of scientists at the 1996 meeting of the American Association for the Advancement of Science, in Baltimore, using a pair of colored jump ropes to demonstrate his mathematical notation for describing -- and untying -- knots.

Handing the ends of the two ropes to four awkward volunteers from his audience, Conway rattled off orders like a caller at a square dance. The dancers obeyed, and in minutes, their ropes were in a complex tangle.

But then almost as quickly, he guided them through the simple mathematical computations and the corresponding movements they needed to undo their knot. And before they knew it, the volunteers and their tethers were freed.

"So you've learned something, and with luck, something with a little bit of joy in it," he says of his shtick. It's a show he has given over and over, delighting and fascinating groups from high school students to Ph.D.s.

Mathematics doesn't have to be dull, he says. Nor do mathematicians.

Conway was born 58 years ago in Liverpool, England, and educated at Cambridge University. But he demolishes any stereotypes you may harbor of the stuffy, distracted British math professor.

He is sturdily built, with a full, gray-flecked beard and light brown hair that hangs to his collar, giving him more the appearance of a graduate student.

Cocky, funny and irreverent, he arrives for breakfast at Baltimore's swank Harbor Court Hotel on a summer morning dressed in short sleeves, short pants and sandals. His pockets are filled with strings and brightly colored plastic jump ropes, which he spreads out on the white linen tablecloth to illustrate his points.

Knot theory got its start in the last century, he says, when British mathematician and physicist Lord Kelvin began wondering whether the mysteries of chemical reactions could be explained by the entanglement of "vortex lines" in the "ether."

He enlisted a colleague named P. G. Tait to study knots to find whether there was anything in them that "looked like chemistry," Conway says.

A decade later, Kelvin's theory of vortex lines had evaporated. But Tait had produced a useful table classifying knots according to the number of times one part of the string crossed another.

At the breakfast table, Conway pushes back the crystal, grabs his red and green jump rope and deftly demonstrates the common overhand knot or trefoil, with three crossings. Then there's the figure-eight knot, with four crossings; the double overhand knot, with five, and the stevedore's knot, with six. There are many more.

"I may be the only person in the world who knows all the knots with seven crossings," he said, with characteristic modesty.

After Tait, he said, a succession of "oddball 19th-century characters" continued to study knots until they had a list of all the possible knots up to 10 crossings.

But why bother?

"Because it's extremely interesting," Conway says. "Suppose you have some knots and they strike you as interesting. And suppose you have a friend in Australia and you need to explain them to that friend over the phone. How do you do it?"

To theorists, knots are usually closed loops in three-dimensional space -- like an extension cord plugged into itself. Knots with loose ends are rarely discussed.

The loops can be highly tangled or linked to other loops, or both. One knot is considered identical to another if it can be deformed to match the other without cutting it or requiring it to pass through itself. But confirming mathematically that a knot can be unraveled and revealed as equivalent to another can be very difficult.

Beyond that, there are "tame" and "wild" knots, "prime" knots and "unknots," "left-handed" and "right-handed" knots. There are even "knot complements" -- all of space except for that occupied by a knot. And some theorists have extended the field into four dimensions, producing three-dimensional color computer graphics to illustrate their work.

Conway bristles when someone asks, as someone always does, whether any of this is good for anything.

"I would not like to sell this on the grounds that the military are interested in it," he says. "The simple fact is that these are interesting."