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knots. This was not a trivial task. You must realize that at every crossing, there are two ways to choose which strand would be uppermost. This means that if a curve contains, say, seven crossings, there are 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 knots to consider. In other words, human life is too short to complete in this intuitive way the classification of knots with tens of crossings or more. Nevertheless, Tait’s labor did not go unappreciated. The great James Clerk Maxwell, who formulated the classical theory of electricity and magnetism, treated Thomson’s atomic theory with respect, stating that “it satisfies more of the conditions than any atom hitherto considered.” Being at the same time well aware of Tait’s contribution, Maxwell offered the following rhyme:

Figure 55

Clear your coil of kinkings

Into perfect plaiting,

Locking loops and linkings

Interpenetrating.

By 1877, Tait had classified alternating knots with up to seven crossings. Alternating knots are those in which the crossings go alternately over and under, like the thread in a woven carpet. Tait also made a few more pragmatic discoveries, in the form of basic principles that were later christened Tait’s conjectures. These conjectures were so substantial, by the way, that they resisted all attempts to prove them rigorously until the late 1980s. In 1885, Tait published tables of knots with up to ten crossings, and he decided to stop there. Independently, University of Nebraska professor Charles Newton Little (1858–1923) also published (in 1899) tables of nonalternating knots with ten or fewer crossings.

Lord Kelvin always thought fondly of Tait. At a ceremony at Peter-house College in Cambridge, where a portrait of Tait was presented, Lord Kelvin said:

I remember Tait once remarking that nothing but science is worth living for. It was sincerely said, but Tait himself proved it to be not true. Tait was a great reader. He would get Shakespeare, Dickens, and Thackeray off by heart. His memory was wonderful. What he once read sympathetically he ever after remembered.

Unfortunately, by the time Tait and Little completed their heroic work on knot tabulation, Kelvin’s theory had already been totally discarded as a potential atomic theory. Still, interest in knots continued for its own sake, the difference being that, as the mathematician Michael Atiyah has put it, “the study of knots became an esoteric branch of pure mathematics.”

The general area of mathematics where qualities such as size, smoothness, and in some sense even shape are ignored is called topology. Topology—the rubber-sheet geometry—examines those properties that remain unchanged when space is stretched or deformed in any fashion (without tearing off pieces or poking holes). By their very nature, knots belong in topology. Incidentally, mathematicians distinguish between knots, which are single knotted loops, links, which are sets of knotted loops all tangled together, and braids, which are sets of vertical strings attached to a horizontal bar at the top and bottom ends.

If you were not impressed with the difficulty of classifying knots, consider the following very telling fact. Charles Little’s table, published in 1899 after six years of work, contained forty-three nonalternating knots of ten crossings. This table was scrutinized by many mathematicians and believed to be correct for seventy-five years. Then in 1974, the New York lawyer and mathematician Kenneth Perko was experimenting with ropes on his living room floor. To his surprise, he discovered that two of the knots in Little’s table were in fact the same. We now believe that there are only forty-two distinct nonalternating knots of ten crossings.

While the twentieth century witnessed great strides in topology, progress in knot theory was relatively slow. One of the key goals of the mathematicians studying knots has been to identify properties that truly distinguish knots. Such properties are called invariants of knots—they represent quantities for which any two different projections of the same knot yield precisely the same