Introduction to Vassiliev Knot Invariants
Cambridge University Press
Edition: Illustrated, 5/24/2012
EAN 9781107020832, ISBN10: 1107020832
Hardcover, 520 pages, 25.4 x 17.8 x 3 cm
Language: English
Originally published in English
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.
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.
'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