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Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134), Volume 134 : Annals of Mathematics Studies (Paperback) - Louis H. Kauffman

Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134), Volume 134

Annals of Mathematics Studies (Paperback)


Published: 25th July 1994
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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.

"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."--Mathematical Reviews

Introductionp. 1
Bracket Polynomial, Temperley-Lieb Algebrap. 5
Jones-Wenzl Projectorsp. 13
The 3-Vertexp. 22
Properties of Projectors and 3-Verticesp. 36
[theta]-Evaluationsp. 45
Recoupling Theory Via Temperley-Lieb Algebrap. 60
Chromatic Evaluations and the Tetrahedronp. 76
A Summary of Recoupling Theoryp. 93
A 3-Manifold Invariant by State Summationp. 102
The Shadow Worldp. 114
The Witten-Reshetikhin-Turaev Invariantp. 129
Blinks [actual symbol not reproducible] 3-Gems: Recognizing 3-Manifoldsp. 160
Tables of Quantum Invariantsp. 185
Bibliographyp. 290
Indexp. 295
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691036403
ISBN-10: 0691036403
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 312
Published: 25th July 1994
Country of Publication: US
Dimensions (cm): 23.44 x 15.52  x 1.98
Weight (kg): 0.41