Introduction to Vassiliev knot invariants:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge University Press
2012
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Cover image Inhaltsverzeichnis |
| Item Description: | "With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced.This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots"-- Provided by publisher. |
| Physical Description: | XVI, 504 S. Ill., graph. Darst. |
| ISBN: | 9781107020832 |
Staff View
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| 245 | 1 | 0 | |a Introduction to Vassiliev knot invariants |c S. Chmutov ; S. Duzhin ; J. Mostovoy |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2012 | |
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| 338 | |b nc |2 rdacarrier | ||
| 500 | |a "With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced.This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots"-- Provided by publisher. | ||
| 650 | 4 | |a Knot theory | |
| 650 | 4 | |a Invariants | |
| 650 | 7 | |a MATHEMATICS / Topology |2 bisacsh | |
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Record in the Search Index
| _version_ | 1804149425499537408 |
|---|---|
| adam_text | Titel: Introduction to Vassiliev knot invariants
Autor: Chmutov, Sergei
Jahr: 2012
Contents
Preface page xi
Knots and their relatives 1
1.1 Definitions and examples 1
1.2 Plane knot diagrams 5
1.3 Inverses and mirror images 7
1.4 Knot tables 9
1.5 Algebra of knots 10
1.6 Tangles, string links and braids 12
1.7 Variations 17
Exercises 21
Knot invariants 26
2.1 Definition and first examples 26
2.2 Linking number 27
2.3 The Conway polynomial 30
2.4 The Jones polynomial 32
2.5 Algebra of knot invariants 35
2.6 Quantum invariants 36
2.7 Two-variable link polynomials 43
Exercises 49
Finite type invariants 57
3.1 Definition of Vassiliev invariants 57
3.2 Algebra of Vassiliev invariants 60
3.3 Vassiliev invariants of degrees 0, 1 and 2 64
3.4 Chord diagrams 66
3.5 Invariants of framed knots 68
3.6 Classical knot polynomials as Vassiliev invariants 70
3.7 Actuality tables 77
viii Contents
3.8 Vassiliev invariants oftangles 79
Exercises 81
4 Chord diagrams 84
4.1 Four- and one-term relations 84
4.2 The Fundamental Theorem 87
4.3 Bialgebras of knots and of Vassiliev knot invariants 89
4.4 Bialgebraof chord diagrams 92
4.5 Bialgebra of weight Systems 98
4.6 Primitive elements in srf 101
4.7 Linear chord diagrams 103
4.8 Intersection graphs 104
Exercises 112
5 Jacobi diagrams 115
5.1 Closed Jacobi diagrams 115
5.2 IHX and AS relations 118
5.3 Isomorphism srf - c o 123
5.4 Product and coproduct in ^ 125
5.5 Primitive subspace of ^ 126
5.6 Open Jacobi diagrams 129
5.7 Linear isomorphism SB - *€ 134
5.8 More on the relation between SS and ^ 140
5.9 The three algebras in small degrees 142
5.10 Jacobi diagrams for tangles 143
5.11 Horizontal chord diagrams 150
Exercises 152
6 Lie algebra weight Systems 157
6.1 Lie algebra weight Systems for the algebra $4 157
6.2 Lie algebra weight Systems for the algebra ^ 169
6.3 Lie algebra weight Systems for the algebra SS 181
6.4 Lie superalgebra weight Systems 187
Exercises 190
7 Algebra of 3-graphs 195
7.1 The space of 3-graphs 195
7.2 Edge multiplication 196
7.3 Vertex multiplication 201
7.4 Action of Y on the primitive space 204
7.5 Lie algebra weight Systems for the algebra Y 206
7.6 Vogel s algebra A 210
Exercises 214
Contents ix
8 The Kontsevich integral 216
8.1 First examples 216
8.2 The construction 219
8.3 Example of calculation 223
8.4 The Kontsevich integral for tangles 225
8.5 Convergence of the integral 227
8.6 Invariance of the integral 229
8.7 Changing the number of critical points 234
8.8 The universal Vassiliev invariant 236
8.9 Symmetries and the group-like property of Z(K) 238
8.10 Towards the combinatorial Kontsevich integral 242
Exercises 244
9 Framed knots and cabling Operations 249
9.1 Framed version of the Kontsevich integral 249
9.2 Cabling Operations 253
9.3 Cabling Operations and the Kontsevich integral 258
9.4 Cablings of the Lie algebra weight Systems 262
Exercises 263
10 The Drinfeld associator 265
10.1 The KZ equation and iterated integrals 265
10.2 Calculation of the KZ Drinfeld associator 275
10.3 Combinatorial construction of the Kontsevich integral 291
10.4 General associators 302
Exercises 308
11 The Kontsevich integral: advanced features 310
11.1 Mutation 310
11.2 Canonical Vassiliev invariants 313
11.3 Wheeling 317
11.4 The unknot and the Hopf link 330
11.5 Rozansky s rationality conjecture 334
Exercises 336
12 Braids and string links 340
12.1 Basics of the theory of nilpotent groups 340
12.2 Vassiliev invariants for free groups 349
12.3 Vassiliev invariants of pure braids 352
12.4 String links as closures of pure braids 357
12.5 Goussarov groups of knots 362
12.6 Goussarov groups of string links 366
x Contents
12.7 Braid invariants as string link invariants 370
Exercises 373
13 Gauss diagrams 375
13.1 The Goussarov theorem 375
13.2 Canonical actuality tables 386
13.3 The Polyak algebra for Virtual knots 387
13.4 Examples of Gauss diagram formulae 391
13.5 The Jones polynomial via Gauss diagrams 399
Exercises 401
14 Miscellany 402
14.1 The Melvin-Morton Conjecture 402
14.2 The Goussarov-Habiro theory revisited 410
14.3 Willerton s fish and bounds for c^ and 73 419
14.4 Bialgebra of graphs 420
14.5 Estimates for the number of Vassiliev knot invariants 424
Exercises 432
15 The space of all knots 434
15.1 The space of all knots 435
15.2 Complements of discriminants 437
15.3 The space of singular knots and Vassiliev invariants 443
15.4 Topology of the diagram complex 448
15.5 Homology of the space of knots and Poisson algebras 454
Appendix 456
A. 1 Lie algebras and their representations 456
A.2 Bialgebras and Hopf algebras 465
A.3 Free algebras and free Lie algebras 480
References 483
Notations 496
Index 499
|
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| author | Chmutov, Sergei 1959- |
| author_GND | (DE-588)1023510367 (DE-588)1023516128 (DE-588)1023064405 |
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| dewey-raw | 514/.2242 |
| dewey-search | 514/.2242 |
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| discipline | Mathematik |
| edition | 1. publ. |
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| id | DE-604.BV040382764 |
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| indexdate | 2024-07-10T00:22:50Z |
| institution | BVB |
| isbn | 9781107020832 |
| language | English |
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| spelling | Chmutov, Sergei 1959- Verfasser (DE-588)1023510367 aut Introduction to Vassiliev knot invariants S. Chmutov ; S. Duzhin ; J. Mostovoy 1. publ. Cambridge [u.a.] Cambridge University Press 2012 XVI, 504 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier "With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced.This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots"-- Provided by publisher. Knot theory Invariants MATHEMATICS / Topology bisacsh Knotentheorie (DE-588)4164318-5 gnd rswk-swf Knotentheorie (DE-588)4164318-5 s DE-604 Dužin, Sergej V. 1956- Sonstige (DE-588)1023516128 oth Mostovoy, Jacob 1970- Sonstige (DE-588)1023064405 oth http://assets.cambridge.org/97811070/20832/cover/9781107020832.jpg Cover image HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025236257&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chmutov, Sergei 1959- Introduction to Vassiliev knot invariants Knot theory Invariants MATHEMATICS / Topology bisacsh Knotentheorie (DE-588)4164318-5 gnd |
| subject_GND | (DE-588)4164318-5 |
| title | Introduction to Vassiliev knot invariants |
| title_auth | Introduction to Vassiliev knot invariants |
| title_exact_search | Introduction to Vassiliev knot invariants |
| title_full | Introduction to Vassiliev knot invariants S. Chmutov ; S. Duzhin ; J. Mostovoy |
| title_fullStr | Introduction to Vassiliev knot invariants S. Chmutov ; S. Duzhin ; J. Mostovoy |
| title_full_unstemmed | Introduction to Vassiliev knot invariants S. Chmutov ; S. Duzhin ; J. Mostovoy |
| title_short | Introduction to Vassiliev knot invariants |
| title_sort | introduction to vassiliev knot invariants |
| topic | Knot theory Invariants MATHEMATICS / Topology bisacsh Knotentheorie (DE-588)4164318-5 gnd |
| topic_facet | Knot theory Invariants MATHEMATICS / Topology Knotentheorie |
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