Introduction to Vassiliev Knot Invariants First draft. Comments welcome. July 20, 2010 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, 191011, Russia E-mail address: [email protected] Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000 Mexico,´ D.F. Mexico E-mail address: [email protected] Contents Preface 8 Part 1. Fundamentals Chapter 1. Knots and their relatives 15 1.1. Definitions and examples 15 § 1.2. Isotopy 16 § 1.3. Plane knot diagrams 19 § 1.4. Inverses and mirror images 21 § 1.5. Knot tables 23 § 1.6. Algebra of knots 25 § 1.7. Tangles, string links and braids 25 § 1.8. Variations 30 § Exercises 34 Chapter 2. Knot invariants 39 2.1. Definition and first examples 39 § 2.2. Linking number 40 § 2.3. Conway polynomial 43 § 2.4. Jones polynomial 45 § 2.5. Algebra of knot invariants 47 § 2.6. Quantum invariants 47 § 2.7. Two-variable link polynomials 55 § Exercises 62 3 4 Contents Chapter 3. Finite type invariants 69 3.1. Definition of Vassiliev invariants 69 § 3.2. Algebra of Vassiliev invariants 72 § 3.3. Vassiliev invariants of degrees 0, 1 and 2 76 § 3.4. Chord diagrams 78 § 3.5. Invariants of framed knots 80 § 3.6. Classical knot polynomials as Vassiliev invariants 82 § 3.7. Actuality tables 88 § 3.8. Vassiliev invariants of tangles 91 § Exercises 93 Chapter 4. Chord diagrams 95 4.1. Four- and one-term relations 95 § 4.2. The Fundamental Theorem 98 § 4.3. Bialgebras of knots and Vassiliev knot invariants 100 § 4.4. Bialgebra of chord diagrams 103 § 4.5. Bialgebra of weight systems 108 § 4.6. Primitive elements in fr 111 § A 4.7. Linear chord diagrams 113 § 4.8. Intersection graphs 114 § Exercises 121 Part 2. Diagrammatic Algebras 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 Contents 5 Chapter 6. Lie algebra weight systems 167 6.1. Lie algebra weight systems for the algebra fr 167 § A 6.2. Lie algebra weight systems for the algebra 179 § C 6.3. Lie algebra weight systems for the algebra 190 § B 6.4. Lie superalgebra weight systems 196 § Exercises 199 Chapter 7. Algebra of 3-graphs 205 7.1. The space of 3-graphs 206 § 7.2. Edge multiplication 206 § 7.3. Vertex multiplication 211 § 7.4. Action of Γ on the primitive space 213 § P 7.5. Lie algebra weight systems for the algebra Γ 215 § 7.6. Vogel’s algebra Λ 220 § Exercises 223 Part 3. The Kontsevich Integral 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. Proof of the Kontsevich theorem 246 § 8.9. Symmetries and the group-like property of Z(K) 248 § 8.10. Towards the combinatorial Kontsevich integral 252 § Exercises 254 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 271 § Exercises 272 6 Contents 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 § 10.4. General associators 310 § Exercises 316 Part 4. Other Topics Chapter 11. The Kontsevich integral: advanced features 321 11.1. Mutation 321 § 11.2. Canonical Vassiliev invariants 324 § 11.3. Wheeling 327 § 11.4. The unknot and the Hopf link 339 § 11.5. Rozansky’s rationality conjecture 344 § Exercises 345 Chapter 12. Braids and string links 351 12.1. Basics of the theory of nilpotent groups 352 § 12.2. Vassiliev invariants for free groups 360 § 12.3. Vassiliev invariants of pure braids 363 § 12.4. String links as closures of pure braids 367 § 12.5. Goussarov groups of knots 372 § 12.6. Goussarov groups of string links 376 § 12.7. Braid invariants as string link invariants 380 § Exercises 383 Chapter 13. Gauss diagrams 385 13.1. The Goussarov theorem 385 § 13.2. Canonical actuality tables 396 § 13.3. The Polyak algebra for virtual knots 397 § 13.4. Examples of Gauss diagram formulae 400 § 13.5. The Jones polynomial via Gauss diagrams 407 § Exercises 409 Chapter 14. Miscellany 411 14.1. The Melvin–Morton Conjecture 411 § 14.2. The Goussarov–Habiro theory revisited 419 § Contents 7 14.3. Willerton’s fish and bounds for c and j 427 § 2 3 14.4. Bialgebra of graphs 428 § 14.5. Estimates for the number of Vassiliev knot invariants 432 § Exercises 440 Chapter 15. The space of all knots 443 15.1. The space of all knots 444 § 15.2. Complements of discriminants 446 § 15.3. The space of singular knots and Vassiliev invariants 452 § 15.4. Topology of the diagram complex 456 § 15.5. Homology of the space of knots and Poisson algebras 462 § Appendix 465 A.1. Lie algebras and their representations 465 § A.2. Bialgebras and Hopf algebras 473 § A.3. Free algebras and free Lie algebras 487 § Bibliography 491 Notations 505 Index 509 8 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 seminar as early as in 1987. Preface 9 Using simplical resolutions, Vassiliev constructs a spectral sequence for the homology of Σ. After applying the Alexander duality this spectral se- quence produces cohomology classes for the space of knots M Σ; in di- − mension zero these are precisely the Vassiliev knot invariants. Vassiliev’s approach, which is technically rather demanding, was sim- plified by J. Birman and X.-S. Lin in [BL]. They explained the relation between the Jones polynomial and finite type invariants and emphasized the role of the algebra of chord diagrams. M. Kontsevich showed that the study of real-valued Vassiliev invariants can, in fact, be reduced entirely to the combinatorics of chord diagrams [Kon1]. His proof used an ana- lytic tool (the Kontsevich integral) which may be thought of as a powerful generalization of the Gauss linking formula. D. Bar-Natan was the first to give a comprehensive treatment of Vassiliev knot and link invariants. In his preprint [BN0] and PhD thesis [BNt] he found the relationship between finite type invariants and the topological quantum field theory developed by his thesis advisor E.