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Introduction to Vassiliev Knot Invariants

  • Authors:
  • S. Chmutov, Ohio State University
  • S. Duzhin, Steklov Institute of Mathematics, St Petersburg
  • J. Mostovoy, Instituto Politécnico Nacional, Mexico
  • Date Published: May 2012
  • availability: Available
  • format: Hardback
  • isbn: 9781107020832


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About the Authors
  • 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.

    • Assumes no prior knowledge of knot theory
    • Includes hundreds of worked examples, illustrations and exercises to suit graduate and undergraduate students
    • Explores connections with graph theory, number theory, Lie algebras, group theory and algebraic topology to help readers understand the theory in context
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: May 2012
    • format: Hardback
    • isbn: 9781107020832
    • length: 520 pages
    • dimensions: 253 x 177 x 30 mm
    • weight: 1.08kg
    • contains: 430 b/w illus. 15 tables 375 exercises
    • availability: Available
  • 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

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    Introduction to Vassiliev Knot Invariants

    S. Chmutov, S. Duzhin, J. Mostovoy

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  • Authors

    S. Chmutov, Ohio State University
    S. Chmutov is Associate Professor in the Department of Mathematics at Ohio State University.

    S. Duzhin, Steklov Institute of Mathematics, St Petersburg
    S. Duzhin is a Senior Researcher in the St Petersburg Department of the Steklov Institute of Mathematics.

    J. Mostovoy, Instituto Politécnico Nacional, Mexico
    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.

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