Temperley-Lieb recoupling theory and invariants of 3-manifolds.

*(English)*Zbl 0821.57003
Annals of Mathematics Studies. 134. Princeton, NJ: Princeton University Press. viii, 296 p. (1994).

New ideas and concepts emerging partly from physics have recently led to spectacular progress in our understanding of low dimensional topology and in particular produced a wealth of new invariants for 3-manifolds some of which might eventually shed some light on the Poincaré conjecture. This prompted a period of intense development which started with the pioneering work of V. Jones, E. Witten, and others and provided new and unexpected interactions between mathematics and physics.

The monograph under review reflects this activity, by presenting in detail several of these developments in some of which the first named author substantially took part himself. It is readable and stimulating and contains new material and material which otherwise is only spread over the literature. Its aim is to develop, in a self-contained manner, a recoupling theory for coloured knots and links with trivalent graphical vertices, based on properties of the bracket polynomial and the tangle- theoretic Temperley-Lieb algebra.

It begins with the bracket polynomial model of the original Jones polynomial and its relation with the Temperley-Lieb algebra. Thereafter Jones-Wenzl projectors in the Temperley-Lieb algebra are introduced as \(q\)-deformed symmetrizers. This construction is conceptually interesting and is computationally useful for the chromatic evaluation method, see below. The authors go on to define the 3-vertex in terms of the mentioned projectors and to discuss certain \(n\)-fold cabled bracket invariants. The following two chapters take up the computation of certain networks at roots of unity and for certain generic values of the argument. Thereafter the recoupling theory is carried out both for generic arguments and at roots of unity. In particular, the orthogonality and the pentagon (Biedenharn-Elliot) identities for the 6j-symbols are derived and network explicit formulas for them are given. Thereafter classical theta and tetrahedral coefficients are computed by means of chromatic evaluation and certain recursion relations for the \(q\)-deformed case are given. After a summary of the recoupling theory developed over the first eight chapters of the monograph, the construction of a state sum 3-manifold invariant relying on recoupling theory is then given. Up to normalization, this yields the Turaev-Viro invariant. Thereafter the shadow world of Kirillov and Reshetikhin is introduced and reformulated in the context of the Temperley-Lieb recoupling theory; within it, coloured link invariants are carried to partition functions of the kind of the Turaev-Viro invariant. Next, a recoupling theory construction of the Witten-Reshetikhin-Turaev invariant is given via surgery and the Kirby calculus, and this invariant is then reformulated as a shadow world partition function on a certain 2-complex. The next to the last chapter contains an algorithm for the computation of the Witten-Reshetikhin- Turaev invariants involving blinks, that is, finite planar graphs with a bipartition on the edges and 3-gems, that is, certain edge coloured graphs that encode 3-manifolds. Finally, the last chapter presents tables of quantum invariants for 3-manifolds which resulted from implementation of the algorithm mentioned before on a computer. These tables also exhibit the classification powers of algorithms designed by the second named author for identifying 3-manifolds from their graphical encodements.

The book should be an essential purchase for mathematics libraries and is likely to be a standard reference for years to come, providing an introduction to an attractive area of further research.

The monograph under review reflects this activity, by presenting in detail several of these developments in some of which the first named author substantially took part himself. It is readable and stimulating and contains new material and material which otherwise is only spread over the literature. Its aim is to develop, in a self-contained manner, a recoupling theory for coloured knots and links with trivalent graphical vertices, based on properties of the bracket polynomial and the tangle- theoretic Temperley-Lieb algebra.

It begins with the bracket polynomial model of the original Jones polynomial and its relation with the Temperley-Lieb algebra. Thereafter Jones-Wenzl projectors in the Temperley-Lieb algebra are introduced as \(q\)-deformed symmetrizers. This construction is conceptually interesting and is computationally useful for the chromatic evaluation method, see below. The authors go on to define the 3-vertex in terms of the mentioned projectors and to discuss certain \(n\)-fold cabled bracket invariants. The following two chapters take up the computation of certain networks at roots of unity and for certain generic values of the argument. Thereafter the recoupling theory is carried out both for generic arguments and at roots of unity. In particular, the orthogonality and the pentagon (Biedenharn-Elliot) identities for the 6j-symbols are derived and network explicit formulas for them are given. Thereafter classical theta and tetrahedral coefficients are computed by means of chromatic evaluation and certain recursion relations for the \(q\)-deformed case are given. After a summary of the recoupling theory developed over the first eight chapters of the monograph, the construction of a state sum 3-manifold invariant relying on recoupling theory is then given. Up to normalization, this yields the Turaev-Viro invariant. Thereafter the shadow world of Kirillov and Reshetikhin is introduced and reformulated in the context of the Temperley-Lieb recoupling theory; within it, coloured link invariants are carried to partition functions of the kind of the Turaev-Viro invariant. Next, a recoupling theory construction of the Witten-Reshetikhin-Turaev invariant is given via surgery and the Kirby calculus, and this invariant is then reformulated as a shadow world partition function on a certain 2-complex. The next to the last chapter contains an algorithm for the computation of the Witten-Reshetikhin- Turaev invariants involving blinks, that is, finite planar graphs with a bipartition on the edges and 3-gems, that is, certain edge coloured graphs that encode 3-manifolds. Finally, the last chapter presents tables of quantum invariants for 3-manifolds which resulted from implementation of the algorithm mentioned before on a computer. These tables also exhibit the classification powers of algorithms designed by the second named author for identifying 3-manifolds from their graphical encodements.

The book should be an essential purchase for mathematics libraries and is likely to be a standard reference for years to come, providing an introduction to an attractive area of further research.

Reviewer: J.Huebschmann (Bonn)

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |