1.4 The Jones polynomial

The Jones polynomial is a polynomial in t, t~l assigned to a knot K in R3. It is denoted by VK(t). It is normalized so that V( t) = 1 for the unknot (the standard unknotted circle in R3). Moreover it has the key property

VK*(t)=VK(tl)           (1.4.1)

where K* is the mirror image of K. Simple examples show that VK(t) need not be invariant under       so that the

Jones polynomial can sometimes distinguish knots from their mirror images. For example the right-handed trefoil knot has

V(t) = t+t3-t4

and so is distinguished from its mirror image. The Alexander polynomial on the other hand always takes the same value for a knot and its mirror image.

The Jones polynomial can be defined (as a Laurent poly­nomial in t1/2) generally, for any oriented link L (i.e. each component of L is oriented). Reversing the orientation of all components leaves the Jones polynomial unchanged. This explains why, for a knot, the orientation is irrelevant.

If we represent a link by a general plane projection with over/under crossings the Jones polynomial can be character­ized and computed by a skein relation. Given any oriented link diagram L, and a crossing point, we can alter the crossing to produce three different diagrams as indicated

 

 

Let V+, V_, V0 denote the Jones polynomials of these links. Then the skein relation is

r1 V+ - tv_ = (tu2 - r1/2) V0.   (1.4.2)

The skein relation is, in a sense, deceptively simple. There is no obvious reason a priori why this relation should define a link invariant: it might depend on the plane presentation.

The way the Jones polynomial was originally discovered was via braids and representations of the Hecke algebra. A braid is a collection of strands as depicted below.

 

 

Note that all strands move upwards. Two braids can be composed in an obvious way, giving the braid group on n strands Bn. Formally we can define Bn as the fundamental group of the configuration space C„ of n distinct points in the plane. The usual picture of a braid can then be viewed asthe space-time graph (with time vertical) of motion along a closed path in C„.

 

Given a braid /3 we can form an oriented link /3 by closing up the braid in a standard way (see below).

 

 

Conjugate elements in Bn give rise to equivalent links. Moreover increasing the number of strands in a braid by a simple twist, as shown, does not affect the corresponding link. A classical theorem of Markov asserts that these two moves generate all equivalences between the resulting links.

Thus to produce an invariant of oriented links one need only produce a class function on all B„ which is unchanged by the move (1.4.3).

Since class functions arise naturally as characters of rep­resentations this suggests we start by considering representa­tions of the braid groups. In fact Jones used representations which came from the Hecke algebra H(n, q). This is the quotient of the group algebra of Bn obtained by requiring the generator <t (a single twist of consecutive strands) to satisfy the quadratic relation

 

 

If q = 1 so that <r2= 1 we get the group algebra of the symmetric group S„. It follows that, for generic values of q, H(n, q) has the same irreducible representations as S„.

For each Young diagram (parametrizing an irreducible representation of S„) we then get a character of Bn which depends on q (as a Laurent polynomial in q1/2). The Jones polynomial (with t = q) is a suitable combination of these characters. In fact only the two-rowed Young diagrams are needed.

The Jones polynomial has been generalized in a variety of ways. One way, described in detail in [17], gives a two-variable polynomial. This also satisfies a skein relation and can be constructed from representations of the Hecke algebra, but now using all Young diagrams.

Another, and more fundamental, way involves choosing a compact Lie group G and an irreducible representation. A polynomial invariant of oriented links is then constructed by using solutions of the Yang-Baxter equations. The original Jones polynomial corresponds to taking G = SU{2) with its standard representation on C2. Taking G = SU(n) for all n, together with their standard representation on Cgives poly­nomials which, taken together, are equivalent to the two- variable polynomial of [17].

Witten's approach, which we shall be describing, also in­volves a choice of group G and a representation. It produces the relevant polynomials in a more direct and natural manner. Moreover, in Witten's theory, we get invariants for links in arbitrary compact 3-manifolds alone (taking the empty link with no components). This is a major advantage and is a convincing demonstration of the naturality of Witten's method.

It is perhaps worth emphasizing that the algebraic or com­binatorial definition of the Jones polynomial is quite elemen­tary and rigorous. It lacks, however, any clear conceptual interpretation. This is precisely what Witten's theory provides, although there are still technical difficulties in developing this side of the theory in all its aspects.

While the Alexander polynomial can be understood in terms of standard algebraic topology (homology theory) and has analogues in higher dimensions, the Jones polynomial is best understood in terms of a purely three-dimensional quantum field theory. There are some indications (discussed in Chapter 6) that the quantum field theory may be related to more standard geometric constructions but this has yet to be worked out.