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An Introduction to Quantum and Vassiliev Knot Invariants

  • David M. Jackson
  • Iain Moffatt
Book

Part of the CMS Books in Mathematics book series (CMSBM)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Basic Knot Theory

    1. Front Matter
      Pages 1-1
    2. David M. Jackson, Iain Moffatt
      Pages 3-22
    3. David M. Jackson, Iain Moffatt
      Pages 23-35
    4. David M. Jackson, Iain Moffatt
      Pages 37-48
    5. David M. Jackson, Iain Moffatt
      Pages 49-59
  3. Quantum Knot Invariants

    1. Front Matter
      Pages 61-61
    2. David M. Jackson, Iain Moffatt
      Pages 63-73
    3. David M. Jackson, Iain Moffatt
      Pages 91-121
    4. David M. Jackson, Iain Moffatt
      Pages 123-148
    5. David M. Jackson, Iain Moffatt
      Pages 149-163
  4. Vassiliev Invariants

    1. Front Matter
      Pages 165-165
    2. David M. Jackson, Iain Moffatt
      Pages 167-179
    3. David M. Jackson, Iain Moffatt
      Pages 181-209
    4. David M. Jackson, Iain Moffatt
      Pages 211-217
    5. David M. Jackson, Iain Moffatt
      Pages 219-248
    6. David M. Jackson, Iain Moffatt
      Pages 249-280
  5. The Kontsevich Invariant

    1. Front Matter
      Pages 281-281
    2. David M. Jackson, Iain Moffatt
      Pages 283-291
    3. David M. Jackson, Iain Moffatt
      Pages 293-308
    4. David M. Jackson, Iain Moffatt
      Pages 309-326
    5. David M. Jackson, Iain Moffatt
      Pages 327-339
  6. Back Matter
    Pages 341-422

About this book

Introduction

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.

Keywords

Vassiliev invariants quantum invariants knots Kontsevich integral Yang-Baxter equation Reshetikhin-Turaev invariant chord diagrams diagrammatic constructions jacobi diagrams quantum groups

Authors and affiliations

  • David M. Jackson
    • 1
  • Iain Moffatt
    • 2
  1. 1.Faculty of MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsRoyal Holloway, University of LondonEghamUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-05213-3
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-05212-6
  • Online ISBN 978-3-030-05213-3
  • Series Print ISSN 1613-5237
  • Series Online ISSN 2197-4152
  • Buy this book on publisher's site