Quantum invariants : a study of knots, 3-manifolds, and their sets / Tomotada Ohtsuki.

Format:

Book

Author/Creator:

Ohtsuki, Tomotada.

Series:

K & E series on knots and everything ; v. 29.

K & E series on knots and everything ; v. 29

Language:

English

Subjects (All):

Quantum field theory.

Knot theory.

Three-manifolds (Topology).

Invariants.

Mathematical physics.

Physical Description:

xiii, 489 pages : illustrations ; 26 cm.

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, [2002]

Summary:

Quantum and related invariants of knots and 3-manifolds are presented, and polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants derived from representation of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. Discussion encompasses the Kontsevich invariants, the theory of Vassiliev invariants, the LMO invariants, and finite type invariants of 3-manifolds. The Chern- Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants. The author is affiliated with the Tokyo Institute of Technology, Japan. Annotation copyrighted by Book News, Inc., Portland, OR.

Contents:

Chapter 1 Knots and polynomial invariants 1

Chapter 2 Braids and representations of the braid groups 23

Chapter 3 Operator invariants of tangles via sliced diagrams 41

Chapter 4 Ribbon Hopf algebras and invariants of links 63

Chapter 5 Monodromy representations of the braid groups derived from the Knizhnik-Zamolodchikov equation 99

Chapter 6 The Kontsevich invariant 133

Chapter 7 Vassiliev invariants 175

Chapter 8 Quantum invariants of 3-manifolds 211

Chapter 9 Perturbative invariants of knots and 3-manifolds 247

Chapter 10 The LMO invariant 269

Chapter 11 Finite type invariants of integral homology 3-spheres 305

Appendix A The quantum group U[subscript q](sl[subscript 2]) 333

Appendix B The quantum sl[subscript 3] invariant via a linear skein 349

Appendix C Braid representations for the Alexander polynomial 357

Appendix D Associators 365

Appendix E Claspers 375

Appendix F Physical background 405

Appendix G Computations for the perturbative invariant 437

Appendix H The quantum sl[subscript 2] invariant and the Kauffman bracket 457.

Notes:

Includes bibliographical references (pages 463-481) and index.

ISBN:

9810246757

OCLC:

49195502