An Introduction to Quantum and Vassiliev Knot Invariants
Authors: Jackson, David M, Moffatt, Iain
Free Preview- Introduces key concepts and constructions both diagrammatic and algebraic in the field
- Exemplifies aspects of problem solving approaches inherent in mathematics
- Demonstrates a range of mathematical concepts tangibly through instantiations in context
- Exposes reader to foundations and applications of mathematical constructions
- Provides exercises throughout text
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- About this book
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This book provides an accessible introduction to knot theory, focussing on Vassiliev invariants, quantum knot invariants constructed via representations of quantum groups, and how these two apparently distinct theories come together through the Kontsevich invariant. Consisting of four parts, the book opens with an introduction to the fundamentals of knot theory, and to knot invariants such as the Jones polynomial. The second part introduces quantum invariants of knots, working constructively from first principles towards the construction of Reshetikhin-Turaev invariants and a description of how these arise through Drinfeld and Jimbo's quantum groups. Its third part offers an introduction to Vassiliev invariants, providing a careful account of how chord diagrams and Jacobi diagrams arise in the theory, and the role that Lie algebras play. The final part of the book introduces the Konstevich invariant. This is a universal quantum invariant and a universal Vassiliev invariant, and brings together these two seemingly different families of knot invariants. The book provides a detailed account of the construction of the Jones polynomial via the quantum groups attached to sl(2), the Vassiliev weight system arising from sl(2), and how these invariants come together through the Kontsevich invariant.
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- Reviews
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“This text is a comprehensive and well written introduction to quantum and Vassiliev invariants of knots. … There is sufficient detail for students and exercises. The text is also an excellent reference for researchers interested in quantum and Vassiliev invariants.” (Heather A. Dye, zbMATH 1425.57007, 2019)
- Table of contents (18 chapters)
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Knots
Pages 3-22
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Knot and Link Invariants
Pages 23-35
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Framed Links
Pages 37-48
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Braids and the Braid Group
Pages 49-59
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R-Matrix Representations of $$\mathfrak {B}_n$$
Pages 63-73
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Table of contents (18 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- An Introduction to Quantum and Vassiliev Knot Invariants
- Authors
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- David M Jackson
- Iain Moffatt
- Series Title
- CMS Books in Mathematics
- Copyright
- 2019
- Publisher
- Springer International Publishing
- Copyright Holder
- Springer Nature Switzerland AG
- eBook ISBN
- 978-3-030-05213-3
- DOI
- 10.1007/978-3-030-05213-3
- Hardcover ISBN
- 978-3-030-05212-6
- Series ISSN
- 1613-5237
- Edition Number
- 1
- Number of Pages
- XX, 422
- Number of Illustrations
- 561 b/w illustrations
- Topics
