An Introduction to Quantum and Vassiliev Knot Invariants

• Main Authors: Jackson, David M. (Author, http://id.loc.gov/vocabulary/relators/aut), Moffatt, Iain (http://id.loc.gov/vocabulary/relators/aut)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2019.
• Edition: 1st ed. 2019.
• Series: CMS Books in Mathematics, Ouvrages de mathématiques de la SMC,
• Subjects: Manifolds (Mathematics).; Complex manifolds.; Nonassociative rings.; Rings (Algebra).; Manifolds and Cell Complexes (incl. Diff.Topology).; Non-associative Rings and Algebras.

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 boo...

Table of Contents:

• Part I Basic Knot Theory
• Knots
• Knot and Link Invariants
• Framed Links
• Braids and the Braid Group
• Part II Quantum Knot Invariants
• R-Matrix Representations of Bn
• Knot Invariants through R-Matrix Representations of Bn
• Operator Invariants
• Ribbon Hopf Algebras
• Reshetikin-Turaev Invariants
• Part III Vassiliev Invarients
• The Fundamentals of Vassiliev Invariants
• Chord Diagrams
• Vassiliev Invariants of Framed Knots
• Jacobi Diagrams
• Lie Algebra Weight Systems
• Part IV The Kontsevich Invariant
• q-tangles
• Jacobi Diagrams on a 1-manifold
• A Construction of the Kontsevich Invariant
• Universality Properties of the Kontsevich Invariant
• Appendix A Background on Modules and Linear Algebra
• Appendix B Rewriting the Definition of Operator Invariants
• Appendix C Computations in Quasi-triangular Hopf Algebras
• Appendix D The Ribbon Hopf Algebra
• Appendix E A Proof of the Invariance of the Reshetikin-Turaev Invariants.