The Jones Polynomial
The theory of the polynomial invented by V. F. R. Jones gives a way of associating to every knot and link a Laurent polynomial with integer coefficients (that is, a finite polynomial expression that can include negative as well as positive powers of the indeterminate). The association of polynomial to link will be made by using a link diagram. The whole theory rests upon the fact that if the diagram is changed by a Reidemeister move, the polynomial stays the same. The polynomial for the link is then defined independently of the choice of diagram. Thus, if two links can be shown, by means of specific calculation from diagrams, to have distinct polynomials, then they are indeed distinct links. This is a relatively easy way of distinguishing knots with diagrams of few crossings. Table 3.1 displays the Jones polynomials for the knots of at most eight crossings shown in Chapter 1. Those polynomials are, by easy inspection, all distinct, so the corresponding knots are all distinct. As will be observed, the Jones polynomial is good, but not infallible, at distinguishing knots. However, that is not its most exciting achievement. Other invariants have, particularly with the aid of computers, always managed to distinguish any interesting pair of knots. Some of those invariants will be encountered in later chapters. The Jones polynomial, however, has been used to prove pleasing new results concerning the possible diagrams that certain knots can possess (see Chapter 5). In addition, the Jones polynomial has been much generalised; it has been developed into a theory, allied in some sense to quantum theory, giving invariants for 3-dimensional manifolds (see Chapter 13) and has been the genesis of a remarkable resurgence of interest in knot theory in all its forms. It is amazing that so simple, powerful and provocative a theory remained unknown until 1984, . Because of the ease with which it can be developed, understood and used, the Jones polynomial has a place very near to the beginning of any exposition of knot theory. The simplest way to define it is by using a slightly different polynomial: the bracket polynomial discovered by L. H. Kauffman .
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