Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134), Volume 134 / Louis H. Kauffman, Sostenes Lins.

Format:

Book

Author/Creator:

Kauffman, Louis H., author.

Lins, Sostenes, author.

Series:

Annals of mathematics studies ; no. 134.

Annals of Mathematics Studies ; 315

Language:

English

Subjects (All):

Knot theory.

Three-manifolds (Topology).

Invariants.

Physical Description:

1 online resource (308 pages) : illustrations.

Place of Publication:

Princeton, NJ : Princeton University Press, [2016]

Language Note:

In English.

Summary:

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.

Contents:

Frontmatter

Contents

Chapter 1. Introduction

Chapter 2. Bracket Polynomial, Temperley-Lieb Algebra

Chapter 3. Jones-Wenzl Projectors

Chapter 4. The 3-Vertex

Chapter 5. Properties of Projectors and 3-Vertices

Chapter 6. θ-Evaluations

Chapter 7. Recoupling Theory Via Temperley-Lieb Algebra

Chapter 8. Chromatic Evaluations and the Tetrahedron

Chapter 9. A Summary of Recoupling Theory

Chapter 10. A 3-Manifold Invariant by State Summation

Chapter 11. The Shadow World

Chapter 12. The Witten-Reshetikhin- Turaev Invariant

Chapter 13. Blinks ↦ 3-Gems: Recognizing 3-Manifolds

Chapter 14. Tables of Quantum Invariants

Bibliography

Index

Notes:

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

9781400882533

1400882532

OCLC:

954123965