By Harold Henderson
Those of us who are neither mathematicians nor poets tend to prefer our paradoxes out of sight and out of mind. But Kauffman seeks them out. Then he scrutinizes them, plays with them, tries to find parallels in other paradoxes. Some of the underlying patterns he discovers will become useful in biology, chemistry, physics. Some will simply remain oddities. Of course you can’t do much math if you worry about which are which, and you can’t tell ahead of time anyway.
The pattern in both the belt trick and the wine dance can be partially described mathematically using the number i, mathematicians’ name for the square root of -1. (Multiply i times itself and you get -1.) “You can think of each half twist [or 180-degree turn] as multiplying by i,” says Kauffman. The belt starts out flat, and he arbitrarily chooses to call that position 1. “Turn it over once. That’s 1 times i, which is the same as just i. Turn it over again. That’s i times i, which is -1.” That’s two twists, or 360 degrees, but you’re not back to where you started. “Turn it over a third time; that’s -1 times i. Turn it over the fourth time [720 degrees]; that’s -1 times i times i, which is -1 times -1, which is 1–where you started.”
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He’s won UIC’s University Scholar award for teaching and research, and he’s held visiting professorships at universities in Swansea, Kyoto, Bologna, Berkeley, and Zaragoza. Along with W.B.R. Lickorish of Cambridge and M. Wadati of the University of Tokyo, he edits the four-year-old Journal of Knot Theory and Its Ramifications. He’s also working on a series of technical books published by Singapore-based World Scientific Publishing, called the “Series on Knots and Everything.” (In what passes for a joke among mathematicians, a colleague told Kauffman that if the books were about everything, then adding “knots” to the title was redundant.) He wrote volume one, the 723-page Knots and Physics; coedited volume three, Quantum Topology; and edited volume six, Knots and Applications.
Knots are beyond ancient. Knot theory is younger than Chicago. It didn’t grow out of mathematical soil first, but out of a seeming paradox that puzzled Sir William Thomson, Baron Kelvin, one of the great minds of 19th-century physics. Physicists back then explained the durability of matter by assuming that it was made up of identical, indestructible, marblelike atoms. But in that case, wondered Kelvin, how do groups of atoms hold together in molecules? And how can gases be compressed and expand?
The belt trick is a blatant paradox. The paradoxes of knots are subtler, perhaps because their patterns are far more intricate. For instance, theorists have long known that there’s only one knot with three crossings, but not until December 1994 did the Journal of Knot Theory publish the news that there are 19,536 different 14-crossing knots that “alternate” (always go over and then under). The “Jones polynomial” excited knot theorists when it was discovered in 1984, because it allowed them to distinguish between many knots and their mirror images–but there are some ten-crossing knots it can’t tell apart, and it’s not yet clear whether it can tell a knot from a mere tangle.
