Temperley-Lieb Recoupling Theory and Invariants of 3-manifolds

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Princeton University Press, 1994 - Mathematics - 296 pages
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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.


 

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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
Bibliography
290
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About the author (1994)

Kauffman-University of Illinois, Chicago

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