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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.
List of contents
| 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 |
About the author
Louis H. Kauffman & Sostenes Lins
Summary
Offers an account of the 3-manifold invariants arising from the original Jones polynomial. This book contains the methods that are based on a recoupling theory for the Temperley-Lieb algebra. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant.
Additional text
"This extremely useful volume provides a self-contained treatment of the construction of 3-manifold invariants directly from the combinatorics of the Jones polynomial in Kauffman's bracket formulation."
