Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134), Volume 134
- 312 pages
- English
- PDF
- Available on iOS & Android
Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134), Volume 134
About This Book
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.
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Table of contents
- Cover
- Title
- Copyright
- Contents
- 1 Introduction
- 2 Bracket Polynomial, Temperley-Lieb Algebra
- 3 Jones-Wenzl Projectors
- 4 The 3-Vertex
- 5 Properties of Projectors and 3-Vertices
- 6 θ-Evaluations
- 7 Recoupling Theory Via Temperley-Lieb Algebra
- 8 Chromatic Evaluations and the Tetrahedron
- 9 A Summary of Recoupling Theory
- 10 A 3-Manifold In variant by State Summation
- 11 The Shadow World
- 12 The Witten-Reshetikhin-Turaev Invariant
- 13 Blinks ⌠3-Gems: Recognizing 3-Manifolds
- 14 Tables of Quantum Invariants
- Bibliography
- Index