Introduction to Vassiliev Knot Invariants

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. Various other topics are then discussed, such as Gauss diagram formulae, before the book ends with Vassiliev's original construction.

Table of Contents:
1. Knots and their relatives; 2. Knot invariants; 3. Finite type invariants; 4. Chord diagrams; 5. Jacobi diagrams; 6. Lie algebra weight systems; 7. Algebra of 3-graphs; 8. The Kontsevich integral; 9. Framed knots and cabling operations; 10. The Drinfeld associator; 11. The Kontsevich integral: advanced features; 12. Braids and string links; 13. Gauss diagrams; 14. Miscellany; 15. The space of all knots; Appendix; References; Notations; Index.

About the Author :
S. Chmutov is Associate Professor in the Department of Mathematics at Ohio State University. S. Duzhin is a Senior Researcher in the St Petersburg Department of the Steklov Institute of Mathematics. J. Mostovoy is Professor in the Department of Mathematics at the Centre for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV-IPN), Mexico City.

Review :
'It is clear that this book is a labour of love, and that no effort has been spared in making it a useful textbook and reference for those seeking to understand its subject. The target readership consists both of first-time learners, for whom pedagogically sound explanations and numerous well-chosen exercises are provided to enhance comprehension, and of experienced mathematicians, for whom many tables of data and readable concise guides to research literature are provided. Numerous figures are included to supplement written explanations. The content is well-modularized, in the sense that different sections of the book may be read independently of one another, and that when there is an essential dependence between sections then this fact is clearly pointed out and the relationship between the sections is explained. This, and a thorough index, combine to make this book not only a valuable textbook, but also a valuable reference.' Zentralblatt MATH 'The book's excellent preface goes on to give an in embryo characterization of the objects in the title … As being a textbook - and an excellent one - the authors take us from a dense but accessible introduction to knots as such to quantum invariants, all in the first two chapters, and then go on to Vassiliev's finite type invariants. Then we get to chord diagrams, Lie algebra connections, Kontsevich's integral, work by Drinfeld, more stuff on the Kontsevich integral, material on braids, and more. The book closes with a chapter on '[t]he space of all knots'. It's very, very attractive material.' Michael Berg, MAA Reviews