## Temperley-Lieb Recoupling Theory and Invariants of 3-manifoldsThis 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. |

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### Contents

Bracket Polynomial TemperleyLieb Algebra | 5 |

JonesWenzl Projectors | 13 |

The 3Vertex | 22 |

Properties of Projectors and 3Vertices | 36 |

Evaluations | 45 |

Recoupling Theory Via TernperleyLieb Algebra | 60 |

Chromatic Evaluations and the Tetrahedron | 76 |

A Summary of Recoupling Theory | 93 |

A 3Manifold Invariant by State Summation | 102 |

The Shadow World | 114 |

The WittenReshetikhinTuraev Invariant | 129 |

Recognizing 3Manifolds | 160 |

Tables of Quantum Invariants | 185 |

290 | |