Overview
- 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
Part of the book series: CMS Books in Mathematics (CMSBM)
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Table of contents (18 chapters)
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Front Matter
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Basic Knot Theory
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Front Matter
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Vassiliev Invariants
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Front Matter
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Keywords
About this book
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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Authors and Affiliations
About the authors
Bibliographic Information
Book Title: An Introduction to Quantum and Vassiliev Knot Invariants
Authors: David M. Jackson, Iain Moffatt
Series Title: CMS Books in Mathematics
DOI: https://doi.org/10.1007/978-3-030-05213-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2019
Hardcover ISBN: 978-3-030-05212-6Published: 16 May 2019
eBook ISBN: 978-3-030-05213-3Published: 04 May 2019
Series ISSN: 1613-5237
Series E-ISSN: 2197-4152
Edition Number: 1
Number of Pages: XX, 422
Number of Illustrations: 561 b/w illustrations
Topics: Manifolds and Cell Complexes (incl. Diff.Topology), Non-associative Rings and Algebras