1.3 History of knot theory

The study of knots (and links) in ordinary three- dimensional space is the archetype of a topological problem. Knots are remarkably complicated things and, even with all the sophisticated techniques of modern topology, they have resisted a definitive treatment. The remarkable developments growing out of the Jones polynomial are an indication of the subtlety of knot theory.

A knot is by definition a smooth-embedding of a circle in R3. Two knots are equivalent if one knot can be deformed continuously into the other without crossing itself. A link is an embedded finite union of disjoint circles.

Knot theory has an interesting history. In the nineteenth century physicists were pondering on the nature of atoms. Lord Kelvin, one of the leading physicists of his time, put forward in 1867 the imaginative and ambitious idea that atoms were knotted vortex tubes of ether [32].

The arguments in favour of this idea may be summarized as follows.

Stability. The stability of matter might be explained by the stability of knots (i.e. their topological nature).

Variety. The variety of chemical elements could be accounted for by the variety of different knots.

Spectrum. Vibrational oscillations of the vortex tubes might explain the spectral lines of atoms.

From a modern twentieth-century point of view we could, in retrospect, have added a fourth.

• (4) Transmutation. The ability of atoms to change into other atoms at high energies could be related to cutting and recombination of knots.

For about 20 years Kelvin's theory of vortex atoms was taken seriously. Maxwell's verdict was that 'it satisfies more of the conditions than any atom hitherto considered'.

Kelvin's collaborator P. G. Tait undertook an extensive study and classification of knots [31]. He enumerated knots in terms of the crossing number of a plane projection and also made some pragmatic discoveries which have since been christened 'Tait's conjectures'. After Kelvin's theory was dis­carded as an atomic theory the study of knots became an esoteric branch of pure mathematics.

Despite the great strides made by topologists in the twen­tieth century the Tait conjectures resisted all attempts to prove them until the late 1980s. The new Jones invariants turned out to be powerful enough to dispose of most of the conjec­tures fairly quickly.

One of the early achievements of modern topology was the discovery in 1928 of the Alexander polynomial of a knot or a link [1]. Although it did not help to prove the Tait conjectures it was an extremely useful knot invariant and greatly simplified the effective classification of knots. The Alexander polynomial arises from the homology of the infinite cyclic cover of the complement of a knot. Equivalently it can be derived from considering cohomology of the knot complement with coefficients in a flat line-bundle. This is very much the context of the Bohm-Aharonov effect.

For more than 50 years the Alexander polynomial remained the only knot invariant of its kind. It was therefore a great surprise to all the experts when, in 1984, Vaughan Jones discovered another polynomial invariant of knots and links. As already mentioned, this turned out to be extremely useful and enabled several of Tait's conjectures to be established.

In the next section we shall briefly summarize some of the key facts about the Jones polynomials. For an excellent and thorough presentation the reader is referred to the account by Jones in [17].