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Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134)
Abstract
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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“The fact that 4d invariance requires the presence of such 'birth' terms has already been argued, on general grounds, in [54].”
From “Sum over surfaces” form of loop quantum gravity (1997) · cites this paper as [54]
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“The precise relation is discussed by Reisenberger [33] and in [18].”
From Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity (1996) · cites this paper as [33]
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Non-Abelian anyons and topological quantum computation
Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Das Sarma, S. (2008) · Reviews of Modern Physics · doi:10.1103/revmodphys.80.1083 · Cited by 6,861
(1998) · doi:10.12942/lrr-1998-1 · Cited by 917
The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677
Topological quantum computation
Freedman, M. H., Kitaev, A., Larsen, M., & Wang, Z. (2002) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-02-00964-3 · Cited by 771
“Sum over surfaces” form of loop quantum gravity
Reisenberger, M. P., & Rovelli, C. (1997) · Physical Review D · doi:10.1103/physrevd.56.3490 · Cited by 372
Metadata
- Authors
- Kauffman, L. H., & Lins, S.
- ORCID
- Kauffman, Louis H.
- Venue
- Princeton University Press eBooks
- Publisher
- Princeton University Press
- Published
- 1994-12-31
- OpenAlex
- W4239309077
- Cited by
- 245
- References
- 0
- Citations by year
- 2026: 5 · 2025: 18 · 2024: 10 · 2023: 15 · 2022: 6 · 2021: 4 · 2020: 7 · 2019: 2 · 2018: 7 · 2017: 12 · 2016: 11 · 2015: 10 · 2014: 9 · 2013: 10 · 2012: 6
- FWCI
- 1.21
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Princeton University Press eBooks (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
- Keywords
- Temperley-Lieb algebra, 3-manifold invariants, Jones polynomial, Reshetikhin-Turaev invariants, Turaev-Viro invariants, Kauffman bracket, 4-manifold invariants
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