3-Manifold Invariants from the Jones Polynomial

Abstract

As proved in Chapter 12, any closed connected orientable 3-manifold can be obtained by the process of surgery on a framed link in S ³. Any invariant of framed links can be applied to such a surgery prescription in the hope of finding an invariant of the 3-manifold. That would need to be some entity associated to the 3-manifold and not just to the particular surgery description; it would need to be unchanged by all possible Kirby moves. An elementary example comes from the idea of linking numbers. A framed link (with components temporarily ordered and oriented) has a linking matrix. This is the symmetric matrix with entries the linking numbers between the pairs of components of the link. The linking number of a component with itself (a diagonal term of the matrix) is taken to be the integer that gives the framing of that component. This linking matrix can easily be seen to be a presentation matrix (in the sense of Chapter 6) for the first homology of the 3-manifold arising from surgery on the framed link. Thus the modulus of the determinant of the matrix, if it is non-zero, is the order of that homology group and the nullity of the matrix is the first Betti number of the manifold. It is easy to check that these numerical invariants do indeed remain unchanged by Kirby moves on the framed link. This, however, is not too exciting, as homology is long and better understood by other means. One might hope to emulate this procedure by a simple direct application of some link invariant. The Alexander polynomial and the Jones polynomial fail in that respect. This chapter explains how the Jones polynomial can nevertheless be amplified to achieve a 3-manifold invariant. Roughly, the idea is to take a linear sum of the Jones polynomials, evaluated at a complex root of unity, of copies of the link with the components replaced by various parallels of the original components. The resulting invariants are known as Witten’s quantum SU _q (2) 3-manifold invariants. The details are somewhat intricate and, as might be expected, will here be eased by the simplifying approach of the Kauffman bracket and the linear skein theory associated with it. The Temperley-Lieb algebras appear as instances of that theory. E. Witten’s initiation of this topic can be found in [135].