Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds

This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose. The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds.

Louis H. Kauffman & Sostenes Lins

1 Introduction 1
2 Bracket Polynomial, Temperley-Lieb Algebra 5
3 Jones-Wenzl Projectors 13
4 The 3-Vertex 22
5 Properties of Projectors and 3-Vertices 36
6 [theta]-Evaluations 45
7 Recoupling Theory Via Temperley-Lieb Algebra 60
8 Chromatic Evaluations and the Tetrahedron 76
9 A Summary of Recoupling Theory 93
10 A 3-Manifold Invariant by State Summation 102
11 The Shadow World 114
12 The Witten-Reshetikhin-Turaev Invariant 129
13 Blinks [actual symbol not reproducible] 3-Gems: Recognizing 3-Manifolds 160
14 Tables of Quantum Invariants 185
Bibliography 290
Index 295

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