• Choose a category
• All
• Classic-Fiction

By Root 706 0

value. In other words, an ideal invariant is literally a “fingerprint” of the knot—a characteristic property of the knot that does not change by deformations of the knot. Perhaps the simplest invariant one can think of is the minimum number of crossings in a drawing of the knot. For instance, no matter how hard you try to disentangle the trefoil knot (figure 54b), you will never reduce the number of crossings to fewer than three. Unfortunately, there are a number of reasons why the minimal number of crossings is not the most useful invariant. First, as figure 55 demonstrates, it is not always easy to determine whether a knot has been drawn with the minimum number of crossings. Second and more important, many knots that are actually different have the same number of crossings. In figure 54, for instance, there are three different knots with six crossings, and no fewer than seven different knots with seven crossings. The minimum number of crossings, therefore, does not distinguish most knots. Finally, the minimum number of crossings, by its very simplistic nature, does not provide much insight into the properties of knots in general.

A breakthrough in knot theory came in 1928 when the American mathematician James Waddell Alexander (1888–1971) discovered an important invariant that has become known as the Alexander polynomial. Basically, the Alexander polynomial is an algebraic expression that uses the arrangement of crossings to label the knot. The good news was that if two knots had different Alexander polynomials, then the knots were definitely different. The bad news was that two knots that had the same polynomial could still be different knots. While extremely helpful, therefore, the Alexander polynomial was still not perfect for distinguishing knots.

Mathematicians spent the next four decades exploring the conceptual basis for the Alexander polynomial and gaining further insights into the properties of knots. Why were they getting so deeply into that subject? Certainly not for any practical application. Thomson’s atomic model had long been forgotten, and there was no other problem in sight in the sciences, economics, architecture, or any other discipline that appeared to require a theory of knots. Mathematicians were spending endless hours on knots simply because they were curious! To these individuals, the idea of understanding knots and the principles that govern them was exquisitely beautiful. The sudden flash of insight afforded by the Alexander polynomial was as irresistible to mathematicians as the challenge of climbing Mount Everest was to George Mallory, who famously replied “Because it is there” to the question of why he wanted to climb the mountain.

In the late 1960s, the prolific English-American mathematician John Horton Conway discovered a procedure for “unknotting” knots gradually, thereby revealing the underlying relationship between knots and their Alexander polynomials. In particular, Conway introduced two simple “surgical” operations that could serve as the basis for defining a knot invariant. Conway’s operations, dubbed flip and smoothing, are described schematically in figure 56. In the flip (figure 56a), the crossing is transformed by running the upper strand under the lower one (the figure also indicates how one might achieve this transformation in a real knot in a string). Note that the flip obviously changes the nature of the knot. For instance, you can easily convince yourself that the trefoil knot in figure 54b would become the unknot (figure 54a) following a flip. Conway’s smoothing operation eliminates the crossing altogether (figure 56b), by reattaching the strands the “wrong” way. Even with the new understanding gained from Conway’s work, mathematicians remained convinced for almost two more decades that no other knot invariants (of the type of the Alexander polynomial) could be found. This situation changed dramatically in 1984.

Figure 56

The New Zealander–American mathematician Vaughan Jones was not studying knots at all. Rather, he was exploring an even more abstract world—one