An Introduction To Quantum And Vassiliev Knot Invariants

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An Introduction to Quantum and Vassiliev Knot Invariants

David M. Jackson,Iain Moffatt
An Introduction to Quantum and Vassiliev Knot Invariants Book PDF

Author : David M. Jackson,Iain Moffatt
Publisher : Springer
Page : 422 pages
File Size : 50,5 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.

Introduction to Vassiliev Knot Invariants

S. Chmutov,Sergeĭ Vasilʹevich Duzhin,J. Mostovoy
Introduction to Vassiliev Knot Invariants Book PDF

Author : S. Chmutov,Sergeĭ Vasilʹevich Duzhin,J. Mostovoy
Publisher : Cambridge University Press
Page : 521 pages
File Size : 49,6 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.

Quantum Invariants

Tomotada Ohtsuki
Quantum Invariants Book PDF

Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 516 pages
File Size : 42,5 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."

Knots and Physics

Louis H. Kauffman
Knots and Physics Book PDF

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 788 pages
File Size : 52,8 Mb
Release : 2001
Category : Science
ISBN : 9789810241117

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Knots and Physics by Louis H. Kauffman Pdf

This invaluable book is an introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes a extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems. In this third edition, a paper by the author entitled "Functional Integration and Vassiliev invariants" has been added. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral. While the book supplies the background, this paper can be read independently as an introduction to quantum field theory and knot invariants and their relation to quantum gravity. As in the second edition, there is a selection of papers by the author at the end of the book. Numerous clarifying remarks have been added to the text.

Quantum Groups and Knot Invariants

Christian Kassel,Marc Rosso,Vladimir G. Turaev
Quantum Groups and Knot Invariants Book PDF

Author : Christian Kassel,Marc Rosso,Vladimir G. Turaev
Publisher : Unknown
Page : 0 pages
File Size : 53,8 Mb
Release : 1997
Category : Categories (Mathematics)
ISBN : 2856290558

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Quantum Groups and Knot Invariants by Christian Kassel,Marc Rosso,Vladimir G. Turaev Pdf

This book provides a concise introduction to quantum groups, braided monoidal categories and quantum invariants of knots and of three-dimensional manifolds. The exposition emphasizes the newly discovered deep relationships between these areas.

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev
Quantum Invariants of Knots and 3 Manifolds Book PDF

Author : Vladimir G. Turaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 608 pages
File Size : 41,7 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

Knot Theory and Its Applications

Kunio Murasugi
Knot Theory and Its Applications Book PDF

Author : Kunio Murasugi
Publisher : Springer Science & Business Media
Page : 348 pages
File Size : 50,9 Mb
Release : 2009-12-29
Category : Mathematics
ISBN : 9780817647193

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Knot Theory and Its Applications by Kunio Murasugi Pdf

This book introduces the study of knots, providing insights into recent applications in DNA research and graph theory. It sets forth fundamental facts such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials. It also covers more recent developments and special topics, such as chord diagrams and covering spaces. The author avoids advanced mathematical terminology and intricate techniques in algebraic topology and group theory. Numerous diagrams and exercises help readers understand and apply the theory. Each chapter includes a supplement with interesting historical and mathematical comments.

Knots and Physics

Louis H Kauffman
Knots and Physics Book PDF

Author : Louis H Kauffman
Publisher : World Scientific
Page : 740 pages
File Size : 46,5 Mb
Release : 1994-01-15
Category : Science
ISBN : 9789814502375

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Knots and Physics by Louis H Kauffman Pdf

In this second edition, the following recent papers have been added: “Gauss Codes, Quantum Groups and Ribbon Hopf Algebras”, “Spin Networks, Topology and Discrete Physics”, “Link Polynomials and a Graphical Calculus” and “Knots Tangles and Electrical Networks”. An appendix with a discussion on invariants of embedded graphs and Vassiliev invariants has also been included. This book is an introduction to knot and link invariants as generalized amplitudes (vacuum–vacuum amplitudes) for a quasi-physical process. The demands of knot theory, coupled with a quantum statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This has the advantage of providing very direct access to the algebra and to the combinatorial topology, as well as the physical ideas. This book is divided into 2 parts: Part I of the book is a systematic course in knots and physics starting from the ground up. Part II is a set of lectures on various topics related to and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics and knots in dynamical systems. Contents:Physical KnotsStates and the Bracket PolynomialThe Jones Polynomial and Its GeneralizationsBraids and the Jones PolynomialFormal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)qYang-Baxter Models for Specializations of the Homfly PolynomialThe Alexander PolynomialKnot-Crystals — Classical Knot Theory in Modern GuiseThe Kauffman PolynomialThree Manifold Invariants from the Jones PolynomialIntegral Heuristics and Witten' s InvariantsThe Chromatic PolynomialThe Potts Model and the Dichromatic PolynomialThe Penrose Theory of Spin NetworksKnots and Strings — Knotted StringsDNA and Quantum Field TheoryKnots in Dynamical Systems — The Lorenz Attractorand other papers Readership: Physicists, mathematical physicists and mathematicians. keywords: Reviews of the First Edition: “It is an attractive book for physicists with profuse and often entertaining illustrations … proofs … seldom heavy and nearly always well explained with pictures… succeeds in infusing his own excitement and enthusiasm for these discoveries and their potential implications.” Physics Today “… here is a gold mine where, with care and patience, one should get acquainted with a beautiful subject under the guidance of a most original and imaginative mind.” Mathematical Reviews

Knots and Physics

Louis H Kauffman
Knots and Physics Book PDF

Author : Louis H Kauffman
Publisher : World Scientific
Page : 864 pages
File Size : 40,5 Mb
Release : 2012-11-09
Category : Science
ISBN : 9789814460309

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Knots and Physics by Louis H Kauffman Pdf

This invaluable book is an introduction to knot and link invariants as generalized amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems. In this new edition, an article on Virtual Knot Theory and Khovanov Homology has beed added. Contents:Physical KnotsStates and the Bracket PolynomialThe Jones Polynomial and Its GeneralizationsBraids and the Jones PolynomialFormal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)qYang–Baxter Models for Specializations of the Homfly PolynomialKnot-Crystals — Classical Knot Theory in a Modern GuiseThe Kauffman PolynomialThree Manifold Invariants from the Jones PolynomialIntegral Heuristics and Witten's InvariantsThe Chromatic PolynomialThe Potts Model and the Dichromatic PolynomialThe Penrose Theory of Spin NetworksKnots and Strings — Knotted StringsDNA and Quantum Field TheoryKnots in Dynamical Systems — The Lorenz Attractorand selected papers Readership: Physicists and mathematicians. Keywords:Knots;Kauffman;Jones PolynomialReviews: "This book is an essential volume for the student of low-dimensional topology from which a serious student can learn most aspects of modern knot theory. Its informal tone encourages investigation on the part of the reader. The author leaves the reader items to puzzle out." Mathematical Reviews Reviews of the Third Edition: “It is an attractive book for physicists with profuse and often entertaining illustrations … proofs … seldom heavy and nearly always well explained with pictures … succeeds in infusing his own excitement and enthusiasm for these discoveries and their potential implications.” Physics Today “The exposition is clear and well illustrated with many examples. The book can be recommended to everyone interested in the connections between physics and topology of knots.” Mathematics Abstracts “… here is a gold mine where, with care and patience, one should get acquainted with a beautiful subject under the guidance of a most original and imaginative mind.” Mathematical Reviews

Knots, Links, Braids and 3-Manifolds

Viktor Vasilʹevich Prasolov,Alekseĭ Bronislavovich Sosinskiĭ
Knots Links Braids and 3 Manifolds Book PDF

Author : Viktor Vasilʹevich Prasolov,Alekseĭ Bronislavovich Sosinskiĭ
Publisher : American Mathematical Soc.
Page : 250 pages
File Size : 54,9 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821808986

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Knots, Links, Braids and 3-Manifolds by Viktor Vasilʹevich Prasolov,Alekseĭ Bronislavovich Sosinskiĭ Pdf

This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. The mathematical prerequisites are minimal compared to other monographs in this area. Numerous figures and problems make this book suitable as a graduate level course text or for self-study.

Quantum Invariants

Tomotada Ohtsuki
Quantum Invariants Book PDF

Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 508 pages
File Size : 44,9 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

The Interface of Knots and Physics

American Mathematical Society
The Interface of Knots and Physics Book PDF

Author : American Mathematical Society
Publisher : American Mathematical Soc.
Page : 221 pages
File Size : 52,6 Mb
Release : 1996
Category : Science
ISBN : 9780821803806

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The Interface of Knots and Physics by American Mathematical Society Pdf

This text is the result of an AMS Short Course on Knots and Physics that was held in San Francisco in January 1994. The authors use ideas and methods of mathematical physics to extract topological information about knots and manifolds. The book features a basic introduction to knot polynomials in relation to statistical link invariants as well as concise introductions to topological quantum field theories and to the role of knot theory in quantum gravity.

Graphs on Surfaces and Their Applications

Sergei K. Lando,Alexander K. Zvonkin
Graphs on Surfaces and Their Applications Book PDF

Author : Sergei K. Lando,Alexander K. Zvonkin
Publisher : Springer Science & Business Media
Page : 455 pages
File Size : 50,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783540383611

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Graphs on Surfaces and Their Applications by Sergei K. Lando,Alexander K. Zvonkin Pdf

Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

Introductory Lectures on Knot Theory

Louis H. Kauffman
Introductory Lectures on Knot Theory Book PDF

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 577 pages
File Size : 46,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814313001

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Introductory Lectures on Knot Theory by Louis H. Kauffman Pdf

More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book.

Quantum Invariants

Tomotada Ohtsuki
Quantum Invariants Book PDF

Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 508 pages
File Size : 40,5 Mb
Release : 2002
Category : Science
ISBN : 9789810246754

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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.