Introduction to Vassiliev knot invariants

512 Pages·2012·5.468 MB·English

by  Chmutov S., Duzhin S., Mostovoy J.

Introduction to Vassiliev Knot Invariants
(aka CDBooK)
— final non-copyedited draft —
S. Chmutov
S. Duzhin
J. Mostovoy
The Ohio State University, Mansfield Campus, 1680 Univer-
sity Drive, Mansfield, OH 44906, USA
E-mail address: [email protected]
Steklov Institute of Mathematics, St. Petersburg Division,
Fontanka 27, St. Petersburg, 191023, Russia
E-mail address: [email protected]
DepartamentodeMatema´ticas, CINVESTAV,ApartadoPostal
14-740, C.P. 07000 Me´xico, D.F. Mexico
E-mail address: [email protected]
This book is published by Cambridge University Press, May 2012, ISBN
978-1-107-02083-2. It incorporates copyediting made by the publisher and
follows the format adopted by CUP. On the other hand, in the present
draft we corrected several typos and inaccuracies that went to the
publisher due to our negligence while reading the proofs. — The authors.
To the memory of V.I.Arnold

Contents
Preface 10
Chapter 1. Knots and their relatives 17
1.1. Definitions and examples 17
§
1.2. Isotopy 18
§
1.3. Plane knot diagrams 21
§
1.4. Inverses and mirror images 23
§
1.5. Knot tables 25
§
1.6. Algebra of knots 27
§
1.7. Tangles, string links and braids 28
§
1.8. Variations 32
§
Exercises 36
Chapter 2. Knot invariants 41
2.1. Definition and first examples 41
§
2.2. Linking number 42
§
2.3. The Conway polynomial 45
§
2.4. The Jones polynomial 47
§
2.5. Algebra of knot invariants 49
§
2.6. Quantum invariants 49
§
2.7. Two-variable link polynomials 57
§
Exercises 64
Chapter 3. Finite type invariants 71
5
6 Contents
3.1. Definition of Vassiliev invariants 71
§
3.2. Algebra of Vassiliev invariants 74
§
3.3. Vassiliev invariants of degrees 0, 1 and 2 78
§
3.4. Chord diagrams 80
§
3.5. Invariants of framed knots 82
§
3.6. Classical knot polynomials as Vassiliev invariants 84
§
3.7. Actuality tables 90
§
3.8. Vassiliev invariants of tangles 93
§
Exercises 95
Chapter 4. Chord diagrams 97
4.1. Four- and one-term relations 97
§
4.2. The Fundamental Theorem 100
§
4.3. Bialgebras of knots and Vassiliev knot invariants 102
§
4.4. Bialgebra of chord diagrams 105
§
4.5. Bialgebra of weight systems 110
§
4.6. Primitive elements in fr 113
§ A
4.7. Linear chord diagrams 115
§
4.8. Intersection graphs 116
§
Exercises 123
Chapter 5. Jacobi diagrams 127
5.1. Closed Jacobi diagrams 127
§
5.2. IHX and AS relations 130
§
5.3. Isomorphism fr 135
§ A ≃ C
5.4. Product and coproduct in 137
§ C
5.5. Primitive subspace of 138
§ C
5.6. Open Jacobi diagrams 141
§
5.7. Linear isomorphism 145
§ B ≃ C
5.8. Relation between and 151
§ B C
5.9. The three algebras in small degrees 153
§
5.10. Jacobi diagrams for tangles 154
§
5.11. Horizontal chord diagrams 161
§
Exercises 163
Chapter 6. Lie algebra weight systems 169
6.1. Lie algebra weight systems for the algebra fr 169
§ A
Contents 7
6.2. Lie algebra weight systems for the algebra 181
§ C
6.3. Lie algebra weight systems for the algebra 192
§ B
6.4. Lie superalgebra weight systems 198
§
Exercises 201
Chapter 7. Algebra of 3-graphs 207
7.1. The space of 3-graphs 208
§
7.2. Edge multiplication 208
§
7.3. Vertex multiplication 213
§
7.4. Action of Γ on the primitive space 215
§ P
7.5. Lie algebra weight systems for the algebra Γ 217
§
7.6. Vogel’s algebra Λ 222
§
Exercises 225
Chapter 8. The Kontsevich integral 227
8.1. First examples 227
§
8.2. The construction 230
§
8.3. Example of calculation 233
§
8.4. The Kontsevich integral for tangles 236
§
8.5. Convergence of the integral 238
§
8.6. Invariance of the integral 239
§
8.7. Changing the number of critical points 245
§
8.8. The universal Vassiliev invariant 246
§
8.9. Symmetries and the group-like property of Z(K) 248
§
8.10. Towards the combinatorial Kontsevich integral 252
§
Exercises 255
Chapter 9. Framed knots and cabling operations 259
9.1. Framed version of the Kontsevich integral 259
§
9.2. Cabling operations 263
§
9.3. The Kontsevich integral of a (p,q)-cable 268
§
9.4. Cablings of the Lie algebra weight systems 272
§
Exercises 273
Chapter 10. The Drinfeld associator 275
10.1. The KZ equation and iterated integrals 275
§
10.2. Calculation of the KZ Drinfeld associator 284
§
10.3. Combinatorial construction of the Kontsevich integral 298
§
8 Contents
10.4. General associators 310
§
Exercises 316
Chapter 11. The Kontsevich integral: advanced features 319
11.1. Mutation 319
§
11.2. Canonical Vassiliev invariants 322
§
11.3. Wheeling 325
§
11.4. The unknot and the Hopf link 337
§
11.5. Rozansky’s rationality conjecture 342
§
Exercises 343
Chapter 12. Braids and string links 347
12.1. Basics of the theory of nilpotent groups 348
§
12.2. Vassiliev invariants for free groups 356
§
12.3. Vassiliev invariants of pure braids 359
§
12.4. String links as closures of pure braids 363
§
12.5. Goussarov groups of knots 368
§
12.6. Goussarov groups of string links 372
§
12.7. Braid invariants as string link invariants 376
§
Exercises 379
Chapter 13. Gauss diagrams 381
13.1. The Goussarov theorem 381
§
13.2. Canonical actuality tables 392
§
13.3. The Polyak algebra for virtual knots 393
§
13.4. Examples of Gauss diagram formulae 396
§
13.5. The Jones polynomial via Gauss diagrams 403
§
Exercises 405
Chapter 14. Miscellany 407
14.1. The Melvin–Morton Conjecture 407
§
14.2. The Goussarov–Habiro theory revisited 415
§
14.3. Willerton’s fish and bounds for c and j 423
2 3
§
14.4. Bialgebra of graphs 424
§
14.5. Estimates for the number of Vassiliev knot invariants 428
§
Exercises 436
Chapter 15. The space of all knots 439
Contents 9
15.1. The space of all knots 440
§
15.2. Complements of discriminants 442
§
15.3. The space of singular knots and Vassiliev invariants 448
§
15.4. Topology of the diagram complex 453
§
15.5. Homology of the space of knots and Poisson algebras 458
§
Appendix 461
A.1. Lie algebras and their representations 461
§
A.2. Bialgebras and Hopf algebras 469
§
A.3. Free algebras and free Lie algebras 483
§
Bibliography 487
Notations 503
10 Contents
Preface
This book is a detailed introduction to the theory of finite type (Vassiliev)
knot invariants, with a stress on its combinatorial aspects. It is intended
to serve both as a textbook for readers with no or little background in this
area, and as a guide to some of the more advanced material. Our aim is
to lead the reader to understanding by means of pictures and calculations,
and for this reason we often prefer to convey the idea of the proof on an
instructive example rather than give a complete argument. While we have
made an effort to make the text reasonably self-contained, an advanced
reader is sometimes referred to the original papers for the technical details
of the proofs.
Historical remarks. The notion of a finite type knot invariant was in-
troduced by Victor Vassiliev (Moscow) in the end of the 1980’s and first
appeared in print in his paper [Va1] (1990). Vassiliev, at the time, was not
specifically interested in low-dimensional topology. His main concern was
the general theory of discriminants in the spaces of smooth maps, and his
description of the space of knots was just one, though the most spectacu-
lar, application of a machinery that worked in many seemingly unrelated
contexts. It was V. I. Arnold [Ar2] who understood the importance of fi-
nite type invariants, coined the name “Vassiliev invariants” and popularized
the concept; since that time, the term “Vassiliev invariants” has become
standard.
A different perspective on the finite type invariants was developed by
Mikhail Goussarov (St.Petersburg). His notion of n-equivalence, which first
appeared in print in [G2] (1993), turned out to be useful in different situa-
tions, for example, in the study of the finite type invariants of 3-manifolds.1
Nowadays some people use the expression “Vassiliev-Goussarov invariants”
for the finite type invariants.
Vassiliev’s definition of finite type invariants is based on the observation
that knots form a topological space and the knot invariants can be thought
of as the locally constant functions on this space. Indeed, the space of knots
is an open subspace of the space M of all smooth maps from S1 to R3; its
complement is the so-called discriminant Σ which consists of all maps that
fail to be embeddings. Two knots are isotopic if and only if they can be
connected in M by a path that does not cross Σ.
1Goussarov cites Vassiliev’s works in his earliest paper [G1]. Nevertheless, according to
O. Viro, Goussarov first mentioned finite type invariants in a talk at the Leningrad topological
seminarasearlyasin1987.