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Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology
V. V. Prasolov, Moscow, Russia, and A. B. Sossinsky, Institute of Electronics and Mathematics, Moscow, Russia

Translations of Mathematical Monographs
1997; 239 pp; softcover
Volume: 154
Reprint/Revision History:
reprinted with corrections 1997; third printing 2001; fourth printing 2012
ISBN-10: 0-8218-0898-2
ISBN-13: 978-0-8218-0898-6
List Price: US$88
Member Price: US$70.40
Order Code: MMONO/154.S
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This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. It emphasizes the geometric aspects of the theory and treats topics such as braids, homeomorphisms of surfaces, surgery of 3-manifolds (Kirby calculus), and branched coverings. This attractive geometric material, interesting in itself yet not previously gathered in book form, constitutes the basis of the last two chapters, where the Jones-Witten invariants are constructed via the rigorous skein algebra approach (mainly due to the Saint Petersburg school).

Unlike several recent monographs, where all of these invariants are introduced by using the sophisticated abstract algebra of quantum groups and representation theory, the mathematical prerequisites are minimal in this book. Numerous figures and problems make it suitable as a course text and for self-study.


Undergraduates, graduate students, research mathematicians and physicists interested in differential geometry.


"Provides an excellent introduction both to classical material and recent developments in 3-dimensional topology and knot theory. The presentation is elementary and extremely clear ... should not be missing in the bookshelf of any working and/or teaching low dimensional topologist."

-- Zentralblatt MATH

"The exposition is excellent throughout. Proofs are clearly illustrated with beautifully drawn diagrams. The text is liberally supplied with exercises, with solutions at the back of the book ... The first part of the book ... would make a good basis for an undergraduate course, and the whole book is just perfect for a graduate course."

-- Bulletin of the London Mathematical Society

"An essentially complete elementary introduction to some important aspects of 3-dimensional topology ... elegantly written with many helpful and well-drawn figures ... the book is self-contained and is accessible to a wide audience including beginning graduate students ... very well written, in a detailed yet comfortable style. I recommend it both to those requiring a foundation for further study of quantum invariants of 3-manifolds and to those wishing a comprehensive introduction to the classical concepts of links and 3-manifolds."

-- Mathematical Reviews

Table of Contents

  • Knots, links, and ribbons
  • Knot and link invariants
  • Braids
  • 3-manifolds
  • Homeomorphisms of surfaces
  • Surgery of 3-manifolds
  • Branched coverings
  • Skein invariants of 3-manifolds
  • Invariants of links in 3-manifolds
  • Appendix
  • Solutions
  • References
  • Index
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