Quantum Invariants

Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 516 pages
File Size : 54,6 Mb
Release : 2002
Category : Invariants
ISBN : 9812811176

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Quantum Invariants by Tomotada Ohtsuki Pdf

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The ChernOCoSimons field theory and the WessOCoZuminoOCoWitten model are described as the physical background of the invariants. Contents: Knots and Polynomial Invariants; Braids and Representations of the Braid Groups; Operator Invariants of Tangles via Sliced Diagrams; Ribbon Hopf Algebras and Invariants of Links; Monodromy Representations of the Braid Groups Derived from the KnizhnikOCoZamolodchikov Equation; The Kontsevich Invariant; Vassiliev Invariants; Quantum Invariants of 3-Manifolds; Perturbative Invariants of Knots and 3-Manifolds; The LMO Invariant; Finite Type Invariants of Integral Homology 3-Spheres. Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics."

Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 508 pages
File Size : 47,6 Mb
Release : 2001-12-21
Category : Mathematics
ISBN : 9789814490719

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Quantum Invariants by Tomotada Ohtsuki Pdf

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern–Simons field theory and the Wess–Zumino–Witten model are described as the physical background of the invariants. Contents: Knots and Polynomial InvariantsBraids and Representations of the Braid GroupsOperator Invariants of Tangles via Sliced DiagramsRibbon Hopf Algebras and Invariants of LinksMonodromy Representations of the Braid Groups Derived from the Knizhnik–Zamolodchikov EquationThe Kontsevich InvariantVassiliev InvariantsQuantum Invariants of 3-ManifoldsPerturbative Invariants of Knots and 3-ManifoldsThe LMO InvariantFinite Type Invariants of Integral Homology 3-Spheres Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics. Keywords:Kontsevich Invariant;LMO Invariant;Quantum Groups;Knot;3-Manifold;Quantum Invariant;Vassiliev Invariant;Finite Type Invariant;Chord Diagram;Jacobi Diagram;KZ Equation;Chern-Simons TheoryReviews:“This is a nicely written and useful book: I think that the author has done a great job in explaining quantum invariants of knots and 3-manifolds also on an intuitive and well-motivated, organically growing and not too technical level, at the same time however presenting a lot of material ordered by a clear guiding line.”Mathematics Abstracts “Ohtsuki's book is a very valuable addition to the literature. It surveys the full spectrum of work in the area of quantum invariants … Ohtsuk's book is very readable, for he makes an attempt to present the material in as straightforward a way as possible … the presentation here is very clear and should be easily accessible … this is an excellent book which I would recommend to beginners wanting to learn about quantum invariants and to experts alike.”Mathematical Reviews

Author : Vladimir G. Turaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 608 pages
File Size : 42,8 Mb
Release : 2016-07-11
Category : Mathematics
ISBN : 9783110435221

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Quantum Invariants of Knots and 3-Manifolds by Vladimir G. Turaev Pdf

Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. Contents:Invariants of graphs in Euclidean 3-space and of closed 3-manifoldsFoundations of topological quantum field theoryThree-dimensional topological quantum field theoryTwo-dimensional modular functors6j-symbolsSimplicial state sums on 3-manifoldsShadows of manifolds and state sums on shadowsConstructions of modular categories

Author : David M. Jackson,Iain Moffatt
Publisher : Springer
Page : 422 pages
File Size : 43,6 Mb
Release : 2019-05-04
Category : Mathematics
ISBN : 9783030052133

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An Introduction to Quantum and Vassiliev Knot Invariants by David M. Jackson,Iain Moffatt Pdf

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.

Author : S. Chmutov,Sergeĭ Vasilʹevich Duzhin,J. Mostovoy
Publisher : Cambridge University Press
Page : 521 pages
File Size : 53,5 Mb
Release : 2012-05-24
Category : Mathematics
ISBN : 9781107020832

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Introduction to Vassiliev Knot Invariants by S. Chmutov,Sergeĭ Vasilʹevich Duzhin,J. Mostovoy Pdf

A detailed exposition of the theory with an emphasis on its combinatorial aspects.

Author : Deyu Tong
Publisher : Unknown
Page : 142 pages
File Size : 52,9 Mb
Release : 1995
Category : Electronic
ISBN : UCAL:C3390779

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Quantum Invariants from Uq(sp(4,C)) by Deyu Tong Pdf

Author : Pieter Cornelis Griend,Pieter van de Griend
Publisher : Unknown
Page : 50 pages
File Size : 46,7 Mb
Release : 1993
Category : Invariants
ISBN : UOM:39015034549587

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Some Background to V. G. Turaev's Quantum Invariants of 3-manifolds by Pieter Cornelis Griend,Pieter van de Griend Pdf

Author : Jeffrey M. Sink
Publisher : Unknown
Page : 260 pages
File Size : 54,6 Mb
Release : 1999
Category : Electronic
ISBN : UOM:39015043222572

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Asymptotic Expansions of Quantum Invariants and a Zeta-function of a Knot by Jeffrey M. Sink Pdf

Author : Thomas Fiedler
Publisher : World Scientific
Page : 341 pages
File Size : 53,7 Mb
Release : 2023-01-04
Category : Mathematics
ISBN : 9789811263019

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One-cocycles And Knot Invariants by Thomas Fiedler Pdf

One-Cocycles and Knot Invariants is about classical knots, i.e., smooth oriented knots in 3-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used to construct combinatorial 1-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and the longitude of the knot. The combinatorial 1-cocycles are therefore lifts of the well-known Conway polynomial of knots, and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.

Author : Louis H. Kauffman,Sostenes Lins
Publisher : Princeton University Press
Page : 312 pages
File Size : 53,9 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882533

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Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134), Volume 134 by Louis H. Kauffman,Sostenes Lins Pdf

This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose. The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds.

Author : J. Scott Carter
Publisher : World Scientific
Page : 398 pages
File Size : 47,6 Mb
Release : 2007
Category : Science
ISBN : 9789812705938

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Intelligence of Low Dimensional Topology 2006 by J. Scott Carter Pdf

This volume gathers the contributions from the international conference ?Intelligence of Low Dimensional Topology 2006,? which took place in Hiroshima in 2006. The aim of this volume is to promote research in low dimensional topology with the focus on knot theory and related topics. The papers include comprehensive reviews and some latest results.

Author : R.B. Sher,R.J. Daverman
Publisher : Elsevier
Page : 1145 pages
File Size : 55,9 Mb
Release : 2001-12-20
Category : Mathematics
ISBN : 9780080532851

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Handbook of Geometric Topology by R.B. Sher,R.J. Daverman Pdf

Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resouce for those already grounded in it, consists of 21 expository surveys written by leading experts and covering active areas of current research. They provide the reader with an up-to-date overview of this flourishing branch of mathematics.

Author : Vassily Olegovich Manturov,Louis H Kauffman
Publisher : World Scientific
Page : 540 pages
File Size : 40,9 Mb
Release : 2015-01-27
Category : Mathematics
ISBN : 9789814630634

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New Ideas In Low Dimensional Topology by Vassily Olegovich Manturov,Louis H Kauffman Pdf

This book consists of a selection of articles devoted to new ideas and developments in low dimensional topology. Low dimensions refer to dimensions three and four for the topology of manifolds and their submanifolds. Thus we have papers related to both manifolds and to knotted submanifolds of dimension one in three (classical knot theory) and two in four (surfaces in four dimensional spaces). Some of the work involves virtual knot theory where the knots are abstractions of classical knots but can be represented by knots embedded in surfaces. This leads both to new interactions with classical topology and to new interactions with essential combinatorics.

Author : Vasilii Olegovich Manturov,Denis Petrovich Ilyutko
Publisher : World Scientific
Page : 553 pages
File Size : 52,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814401135

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Virtual Knots by Vasilii Olegovich Manturov,Denis Petrovich Ilyutko Pdf

The book is the first systematic research completely devoted to a comprehensive study of virtual knots and classical knots as its integral part. The book is self-contained and contains up-to-date exposition of the key aspects of virtual (and classical) knot theory.Virtual knots were discovered by Louis Kauffman in 1996. When virtual knot theory arose, it became clear that classical knot theory was a small integral part of a larger theory, and studying properties of virtual knots helped one understand better some aspects of classical knot theory and encouraged the study of further problems. Virtual knot theory finds its applications in classical knot theory. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary three-manifold and classical knot theory.In this book we present the latest achievements in virtual knot theory including Khovanov homology theory and parity theory due to V O Manturov and graph-link theory due to both authors. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams.Graph-links can be treated as OC diagramless knot theoryOCO: such OC linksOCO have crossings, but they do not have arcs connecting these crossings. It turns out, however, that to graph-links one can extend many methods of classical and virtual knot theories, in particular, the Khovanov homology and the parity theory.

Author : Vladimir Turaev
Publisher : Unknown
Page : 128 pages
File Size : 45,9 Mb
Release : 1994
Category : Electronic
ISBN : OCLC:803451861

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Quantum Invariants of Knots and 3-manifolds by Vladimir Turaev Pdf