Temperley - Lieb Recoupling Theory and Invariants of 3-Manifolds

Abstract

I used to keep the entire spin networks literature in a small folder on my shelf. The recent explosion of interest in the subject has made this impossible; more is probably now written every week than the contents of my folder in 1991. This brings the need to consolidate the subject and collect together results and formulae in one coherent place. Kauffman and Lins' book is a splendid contribution, as the well thumbed reference copy can testify. The renewed interest in the subject came from two areas. Firstly there was the q-revolution which spawned the representation theory of the q-deformation of SU(2) and its application to 3-manifold theory by combinatorial techniques. Secondly, spin networks came to be applied in quantum field theories in places where discrete methods began to seem more fundamental. There are many overlaps between these, and many of the tools developed for specific 3-manifold purposes have, in fact, a much wider application. Perhaps recognizing this, the authors made the first half of the book a review of the recoupling theory of SU(2), leaving the application to 3-manifolds strictly to the second part. The development of the recoupling theory makes no reference to SU(2) itself, as the invariant theory can be developed entirely by the means of diagrams. This is a continuation of the theme of the spin networks and strand networks developed by Penrose, extended to the q-deformed case. The use of diagrams was originally conceived as an aid to understanding algebraic formulae; now the connection runs deeper. We now understand that a planar diagram for the recoupling of representations of SU(2) is an indispensible part of the theory. Without it, the fine detail of the formulae become impossible to keep track of in a coherent way. Thus basing the theory on diagrams is probably the most useful and certainly the quickest way of grasping the subject. The book starts with the Jones - Wenzl projectors, which are the diagrammatic version of the projection onto irreducible representations of SU(2). Then it quickly moves on to calculating the value of q-spin networks with these irreducible representations on the edges. Particular networks are calculated in detail, namely the diagram and the tetrahedron. This then allows the properties of the 6j symbol to be developed. Having all the formulae in one place, and they have proved remarkably reliable, makes this book invaluable, as the literature is plagued by slightly different versions of the same formulae caused by differences in convention. One unfortunate departure from the conventions established in the physics literature is that the curly bracket symbol for the 6j symbol has been assigned a different normalization, which makes the formulae related to associativity simple but not symmetrical under permutations of the tetrahedron. Most of the recoupling theory is developed for general q, and the particular case of q as a root of unity. However, the reader interested only in classical spin networks need not be lost; one can put q = 1 everywhere and still enjoy reading the book. The calculations of q-spin network evaluations are long but tantalizingly similar to the formulae for the classical q = 1 case. General formulae are known only for the classical case. Chapter 8 describes the chromatic method for q = 1 due to Penrose, leaving the generalization to the q-deformation as an unsolved problem. The combinatorics of the chromatic method are somewhat delicate to describe, and the informal method of presentation, which serves the book well elsewhere, is stretched to the limit at this point. The second part of the book is much more specialized. There are reviews of the Turaev - Viro invariant, the Witten - Reshetikhin - Turaev invariant and the `shadow world' description of graph invariants due to Kirillov and Resheatikhin. All of these have been condensed and simplified from their original presentations, but some of the theory, for example, of the topology, has been omitted. The culmination of the book is a series of tables of the WRT invariants for some 3-manifolds, based on the formulae given in the first part of the book.