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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.
Louis H. Kauffman & Sostenes Lins
1Introduction12Bracket Polynomial, Temperley-Lieb Algebra53Jones-Wenzl Projectors134The 3-Vertex225Properties of Projectors and 3-Vertices366[theta]--Evaluations457Recoupling Theory Via Temperley-Lieb Algebra608Chromatic Evaluations and the Tetrahedron769A Summary of Recoupling Theory9310A 3-Manifold Invariant by State Summation10211The Shadow World11412The Witten-Reshetikhin-Turaev Invariant12913Blinks [actual symbol not reproducible] 3-Gems: Recognizing 3-Manifolds16014Tables of Quantum Invariants185Bibliography290Index295
