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An introduction to quantum and Vassiliev knot invariants / David M. Jackson and Iain Moffatt.

Jackson, David M., author.; Moffatt, Iain, author.
2019
QC174.52.K56
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Title
An introduction to quantum and Vassiliev knot invariants / David M. Jackson and Iain Moffatt.
Author
Jackson, David M., author.
ISBN
9783030052133 (electronic book)
3030052133 (electronic book)
9783030052126
DOI
https://doi.org/10.1007/978-3-030-05213-3
Published
Cham, Switzerland : Springer, [2019]
Language
English
Description
1 online resource.
Item Number
10.1007/978-3-030-05213-3 doi
Call Number
QC174.52.K56
Dewey Decimal Classification
530.143
Summary
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.
Bibliography, etc. Note
Includes bibliographical references and index.
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Access limited to authorized users.
Digital File Characteristics
text file PDF
Source of Description
Online resource; title from PDF title page (viewed May 7, 2019).
Added Author
Moffatt, Iain, author.
Series
CMS books in mathematics.
Available in Other Form
Print version: 9783030052126
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Intro; Preface; Contents; Frequently Used Notation; Introduction; Part I Basic Knot Theory; 1 Knots; 1.1 Knots and Equivalence; 1.2 Knot and Link Diagrams; 1.3 The Reidemeister Moves; 1.4 A Proof of Reidemeister's Theorem; 1.5 Oriented Links; 2 Knot and Link Invariants; 2.1 The Idea of an Invariant; 2.2 Skein Relations and Polynomial Invariants; 3 Framed Links; 3.1 Framed Links and Their Diagrams; 3.2 Framed Link Invariants; 3.3 Deframing a Link Invariant; 4 Braids and the Braid Group; 4.1 Braids and Braid Equivalence; 4.2 The Braid Group mathfrakBn; 4.3 The Markov Moves

8.4.1 Lie Algebras8.4.2 The Quantized Universal Enveloping Algebra Uh(mathfraksl2); 8.4.3 The Quantized Universal Enveloping Algebra Uq(mathfraksl2); 8.4.4 Representations; 9 Reshetikhin-Turaev Invariants; 9.1 Coloured Tangles; 9.2 Construction of the Invariants; 9.3 QmathfrakA is an Operator Invariant; 9.4 A Knot Invariant from Uq(mathfraksl2); Part III Vassiliev Invariants; 10 The Fundamentals of Vassiliev Invariants; 10.1 Vassiliev Invariants and Singular Knots; 10.2 Examples of Vassiliev Invariants; 10.3 The Vector Space of Vassiliev Invariants; 11 Chord Diagrams

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