An Introduction to Quantum and Vassiliev Knot Invariants

Description

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. Front Matter ....Pages i-xx Front Matter ....Pages 1-1 Knots (David M. Jackson, Iain Moffatt)....Pages 3-22 Knot and Link Invariants (David M. Jackson, Iain Moffatt)....Pages 23-35 Framed Links (David M. Jackson, Iain Moffatt)....Pages 37-48 Braids and the Braid Group (David M. Jackson, Iain Moffatt)....Pages 49-59 Front Matter ....Pages 61-61 R-Matrix Representations of \(\mathfrak {B}_n\) (David M. Jackson, Iain Moffatt)....Pages 63-73 Knot Invariants Through R-Matrix Representations of \(\mathfrak {B}_n\) (David M. Jackson, Iain Moffatt)....Pages 75-90 Operator Invariants (David M. Jackson, Iain Moffatt)....Pages 91-121 Ribbon Hopf Algebras (David M. Jackson, Iain Moffatt)....Pages 123-148 Reshetikhin–Turaev Invariants (David M. Jackson, Iain Moffatt)....Pages 149-163 Front Matter ....Pages 165-165 The Fundamentals of Vassiliev Invariants (David M. Jackson, Iain Moffatt)....Pages 167-179 Chord Diagrams (David M. Jackson, Iain Moffatt)....Pages 181-209 Vassiliev Invariants of Framed Knots (David M. Jackson, Iain Moffatt)....Pages 211-217 Jacobi Diagrams (David M. Jackson, Iain Moffatt)....Pages 219-248 Lie Algebra Weight Systems (David M. Jackson, Iain Moffatt)....Pages 249-280 Front Matter ....Pages 281-281 q-Tangles (David M. Jackson, Iain Moffatt)....Pages 283-291 Jacobi Diagrams on a 1-Manifold (David M. Jackson, Iain Moffatt)....Pages 293-308 A Construction of the Kontsevich Invariant (David M. Jackson, Iain Moffatt)....Pages 309-326 Universality Properties of the Kontsevich Invariant (David M. Jackson, Iain Moffatt)....Pages 327-339 Back Matter ....Pages 341-422