KNOTS - RECENT DEVELOPMENTS - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER IX. RECENT DEVELOPMENTS

§8. KNOTS

The theory of knots is currently the focus of a tremendous amount of research activity, sparked by the discovery of the Jones Polynomial, a remarkable new method for distinguishing topologically inequivalent knots. The theory involves links as well as knots, and we begin by making these concepts more precise.

A link is a set of one or more closed loops in three-dimensional space. The individual loops are called the components of the link. The loops can be twisted or knotted, and—as the name suggests—may be linked together in any way, including not being linked at all in the usual sense. If there is only one loop, the link is called a knot. The central problem in link theory is to find efficient ways to tell whether or not two given links or knots are topologically equivalent—that is, can be deformed into each other by continuous transformations (see pp. 241–2). In particular we want to find out whether what looks like a knot is really unknotted, that is, is equivalent to the unknot (Fig. 292a); and whether a given n-component link can be unlinked, that is, is equivalent to the n-component unlink (Fig. 292b).

The way to achieve this is to find topological invariants. These are numbers—or more complicated mathematical objects—that do not change when the link is continuously deformed. It follows that link with different invariants must be topologically inequivalent. However, links with the same invariants may or may not be equivalent, and the only way to decide is either to find a topological equivalence or invent a more sensitive invariant.

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Fig. 290. Two copies of a Cantor set triple its size

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Fig. 291. Three copies of a Sierpinski gasket double its size.

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Fig. 292. (a) The unknot. (b) The n-component unlink.

The standard knot invariant, in the pre-Jones era of knot theory, was the Alexander polynomial, invented in 1926. This assigns to each knot a polynomial in a variable t, which can be calculated by following a standard procedure. The precise procedure need not concern us here, but to indicate the kind of results that are obtained, Fig. 293 shows several simple knots and their Alexander polynomials.

The Alexander polynomial is good enough to distinguish between a trefoil knot and a reef knot, because these have different Alexander polynomials. It is not good enough to distinguish between

• a reef knot and a granny knot

• a left-handed trefoil and a right-handed trefoil

even though it is experimentally “obvious” that these knots are indeed inequivalent. The problem is, how can we prove this? Between 1926 and 1984, mathematicians expended a great deal of effort on these and similar questions. They solved them, but by rather complicated methods. Knot theory did not exactly grind to a halt, but it was certainly in need of some new insights.

In 1984 Vaughan Jones, a New Zealander, was working on questions in analysis, about so-called trace functions on operator algebras, which had arisen in connection with mathematical physics. D. Hatt and Pierre de la Harpe noticed that some of his equations looked rather like equations arising in the theory of braids, which are tangled systems of lines very closely related to links. Pondering the reasons that might lie behind such a coincidence, Jones discovered that his trace functions could be used to define a polynomial invariant for links.

At first it was thought that the Jones polynomial must be just some variation on the Alexander polynomial, but it soon became clear that it was genuinely new. Simpler definitions, not involving operator algebras, were found. Five separate groups of mathematicians independently and simultaneously discovered a generalization which was even better at distinguishing knots, a two-variable formula often called the HOMFLY polynomial—short for its discoverers: Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter. Today there are a dozen or more new knot polynomials. They have solved many outstanding problems, but they also pose many new puzzles of their own, because they do not fit comfortably into the established machinery of topology. In a sense, although topologists can calculate them and prove theorems about them, they are not yet certain what these new polynomial invariants really are. They appear to have some deep relation to quantum physics.

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Fig. 293. Some common knots and their Alexander polynomials.

The original Jones polynomial is a powerful enough invariant to distinguish a left-handed trefoil from a right-handed one, which the Alexander polynomial could not manage. The HOMFLY polynomial is even more powerful, and it can distinguish a reef knot from a granny knot. In fact, denoting the HOMFLY polynomial of a link L by P(L), we have

P(left-handed trefoil) = –2x2x4 + x2y2.

P(right-handed trefoil) = –2x–2x–4 + x–2y2.

P(reef) = (–2x2x4 + x2y2)(–2x–2x–4 + x–2y2).

P(granny) = (–2x2x4 + x2y2)2.

Here x and y are the two variables required to define the polynomial. These results obviously prove not only that the two types of trefoil are topologically inequivalent, but also that the reef knot and granny knot are topologically inequivalent.