Ulteriori informazioni
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.
Sommario
| 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 |
Info autore
Louis H. Kauffman is Professor of Mathematics at the University of Illinois, Chicago.
Sostenes Lins is Professor of Mathematics at the Universidade Federal de Pernambuco in Recife, Brazil.
Riassunto
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.
Testo aggiuntivo
"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."
