Quantum invariants of knots and 3-manifolds:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u.a.
De Gruyter
1994
|
| Series: | De Gruyter studies in mathematics
18 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 588 S. graph. Darst. |
| ISBN: | 3110137046 |
Staff View
MARC
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| 100 | 1 | |a Turaev, Vladimir G. |d 1954- |e Verfasser |0 (DE-588)122717791 |4 aut | |
| 245 | 1 | 0 | |a Quantum invariants of knots and 3-manifolds |c V. G. Turaev |
| 264 | 1 | |a Berlin u.a. |b De Gruyter |c 1994 | |
| 300 | |a X, 588 S. |b graph. Darst. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction 1
Part I. Towards Topological Field Theory 15
Chapter I. Invariants of graphs in Euclidean 3 space 17
1. Ribbon categories 17
2. Operator invariants of ribbon graphs 30
3. Reduction of Theorem 2.5 to lemmas 49
4. Proof of lemmas 57
Notes 71
Chapter II. Invariants of closed 3 manifolds 72
1. Modular tensor categories 72
2. Invariants of 3 manifolds 78
3. Proof of Theorem 2.3.2. Action of SL(2, Z) 84
4. Computations in semisimple categories 99
5. Hermitian and unitary categories 108
Notes 116
Chapter III. Foundations of topological quantum field theory 118
1. Axiomatic definition of TQFT s 118
2. Fundamental properties 127
3. Isomorphisms of TQFT s 132
4. Quantum invariants 136
5. Hermitian and unitary TQFT s 142
6. Elimination of anomalies 145
Notes 150
Chapter IV. Three dimensional topological quantum field theory 152
1. Three dimensional TQFT: preliminary version 152
2. Proof of Theorem 1.9 162
3. Lagrangian relations and Maslov indices 179
4. Computation of anomalies 186
viii Contents
5. Action of the modular groupoid 190
6. Renormalized 3 dimensional TQFT 196
7. Computations in the renormalized TQFT 207
8. Absolute anomaly free TQFT 210
9. Anomaly free TQFT 213
10. Hermitian TQFT 217
11. Unitary TQFT 223
12. Verlinde algebra 226
Notes 234
Chapter V. Two dimensional modular functors 236
1. Axioms for a 2 dimensional modular functor 236
2. Underlying ribbon category 247
3. Weak and mirror modular functors 266
4. Construction of modular functors 268
5. Construction of modular functors continued 274
Notes 297
Part II. The Shadow World 299
Chapter VI. 6j symbols 301
1. Algebraic approach to 6j symbols 301
2. Unimodal categories 310
3. Symmetrized multiplicity modules 312
4. Framed graphs 318
5. Geometric approach to 6y symbols 331
Notes 343
Chapter VII. Simplicial state sums on 3 manifolds 345
1. State sum models on triangulated 3 manifolds 345
2. Proof of Theorems 1.4 and 1.7 351
3. Simplicial 3 dimensional TQFT 356
4. Comparison of two approaches 361
Notes 365
Chapter VIII. Generalities on shadows 367
1. Definition of shadows 367
2. Miscellaneous definitions and constructions 371
3. Shadow links 376
Contents ix
4. Surgeries on shadows 382
5. Bilinear forms of shadows 386
6. Integer shadows 388
7. Shadow graphs 391
Notes 393
Chapter IX. Shadows of manifolds 394
1. Shadows of 4 manifolds 394
2. Shadows of 3 manifolds 400
3. Shadows of links in 3 manifolds 405
4. Shadows of 4 manifolds via handle decompositions 410
5. Comparison of bilinear forms 413
6. Thickening of shadows 417
7. Proof of Theorems 1.5 and 1.7 1.11 427
8. Shadows of framed graphs 431
Notes 434
Chapter X. State sums on shadows 435
1. State sum models on shadowed polyhedra 435
2. State sum invariants of shadows 444
3. Invariants of 3 manifolds from the shadow viewpoint 450
4. Reduction of Theorem 3.3 to a lemma 452
5. Passage to the shadow world 455
6. Proof of Theorem 5.6 463
7. Invariants of framed graphs from the shadow viewpoint 473
8. Proof of Theorem VII.4.2 477
9. Computations for graph manifolds 484
Notes 489
Part III. Towards Modular Categories 491
Chapter XI. An algebraic construction of modular categories 493
1. Hopf algebras and categories of representations 493
2. Quasitriangular Hopf algebras 496
3. Ribbon Hopf algebras 500
4. Digression on quasimodular categories 503
5. Modular Hopf algebras 506
6. Quantum groups at roots of unity 508
7. Quantum groups with generic parameter 513
Notes 517
x Contents
Chapter XII. A geometric construction of modular categories 518
1. Skein modules and the Jones polynomial 518
2. Skein category 523
3. The Temperley Lieb algebra 526
4. The Jones Wenzl idempotents 529
5. The matrix 5 535
6. Refined skein category 539
7. Modular and semisimple skein categories 546
8. Multiplicity modules 551
9. Hermitian and unitary skein categories 557
Notes 559
Appendix I. Dimension and trace re examined 561
Appendix II. Vertex models on link diagrams 563
Appendix III. Gluing re examined 565
Appendix IV. The signature of closed 4 manifolds from a state sum 568
Problems 571
References 573
Subject index 585
|
| any_adam_object | 1 |
| author | Turaev, Vladimir G. 1954- |
| author_GND | (DE-588)122717791 |
| author_facet | Turaev, Vladimir G. 1954- |
| author_role | aut |
| author_sort | Turaev, Vladimir G. 1954- |
| author_variant | v g t vg vgt |
| building | Verbundindex |
| bvnumber | BV009707081 |
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| classification_tum | PHY 023f PHY 011f MAT 572f |
| ctrlnum | (OCoLC)231622168 (DE-599)BVBBV009707081 |
| discipline | Physik Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:39:34Z |
| institution | BVB |
| isbn | 3110137046 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006420841 |
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| physical | X, 588 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Turaev, Vladimir G. 1954- Verfasser (DE-588)122717791 aut Quantum invariants of knots and 3-manifolds V. G. Turaev Berlin u.a. De Gruyter 1994 X, 588 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 18 Topologische Invariante (DE-588)4310559-2 gnd rswk-swf Dimension 3 (DE-588)4321722-9 gnd rswk-swf Knoten Mathematik (DE-588)4164314-8 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Monoidale Kategorie (DE-588)4170466-6 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Knotentheorie (DE-588)4164318-5 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 s Topologische Mannigfaltigkeit (DE-588)4185712-4 s Knotentheorie (DE-588)4164318-5 s DE-604 Topologie (DE-588)4060425-1 s Monoidale Kategorie (DE-588)4170466-6 s Mannigfaltigkeit (DE-588)4037379-4 s Dimension 3 (DE-588)4321722-9 s Knoten Mathematik (DE-588)4164314-8 s Topologische Invariante (DE-588)4310559-2 s De Gruyter studies in mathematics 18 (DE-604)BV000005407 18 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006420841&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Turaev, Vladimir G. 1954- Quantum invariants of knots and 3-manifolds De Gruyter studies in mathematics Topologische Invariante (DE-588)4310559-2 gnd Dimension 3 (DE-588)4321722-9 gnd Knoten Mathematik (DE-588)4164314-8 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Monoidale Kategorie (DE-588)4170466-6 gnd Topologie (DE-588)4060425-1 gnd Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Knotentheorie (DE-588)4164318-5 gnd |
| subject_GND | (DE-588)4310559-2 (DE-588)4321722-9 (DE-588)4164314-8 (DE-588)4047984-5 (DE-588)4170466-6 (DE-588)4060425-1 (DE-588)4185712-4 (DE-588)4037379-4 (DE-588)4164318-5 |
| title | Quantum invariants of knots and 3-manifolds |
| title_auth | Quantum invariants of knots and 3-manifolds |
| title_exact_search | Quantum invariants of knots and 3-manifolds |
| title_full | Quantum invariants of knots and 3-manifolds V. G. Turaev |
| title_fullStr | Quantum invariants of knots and 3-manifolds V. G. Turaev |
| title_full_unstemmed | Quantum invariants of knots and 3-manifolds V. G. Turaev |
| title_short | Quantum invariants of knots and 3-manifolds |
| title_sort | quantum invariants of knots and 3 manifolds |
| topic | Topologische Invariante (DE-588)4310559-2 gnd Dimension 3 (DE-588)4321722-9 gnd Knoten Mathematik (DE-588)4164314-8 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Monoidale Kategorie (DE-588)4170466-6 gnd Topologie (DE-588)4060425-1 gnd Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Knotentheorie (DE-588)4164318-5 gnd |
| topic_facet | Topologische Invariante Dimension 3 Knoten Mathematik Quantenfeldtheorie Monoidale Kategorie Topologie Topologische Mannigfaltigkeit Mannigfaltigkeit Knotentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006420841&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
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