Invariants of Knots and 3-manifolds (Kyoto 2001)
Author | : Tomotada Ohtsuki |
Publisher | : |
Total Pages | : 600 |
Release | : 2002 |
Genre | : Knot theory |
ISBN | : |
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Author | : Tomotada Ohtsuki |
Publisher | : |
Total Pages | : 600 |
Release | : 2002 |
Genre | : Knot theory |
ISBN | : |
Author | : Tomotada Ohtsuki |
Publisher | : |
Total Pages | : 572 |
Release | : 2004 |
Genre | : |
ISBN | : |
Author | : Vladimir G. Turaev |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 608 |
Release | : 2016-07-11 |
Genre | : Mathematics |
ISBN | : 3110435225 |
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 | : |
Publisher | : |
Total Pages | : 590 |
Release | : 2002 |
Genre | : Knot theory |
ISBN | : |
Author | : S. Chmutov |
Publisher | : Cambridge University Press |
Total Pages | : 521 |
Release | : 2012-05-24 |
Genre | : Mathematics |
ISBN | : 1107020832 |
A detailed exposition of the theory with an emphasis on its combinatorial aspects.
Author | : Tomotada Ohtsuki |
Publisher | : World Scientific |
Total Pages | : 508 |
Release | : 2001-12-21 |
Genre | : Mathematics |
ISBN | : 9814490717 |
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.
Author | : Viktor Vasilʹevich Prasolov |
Publisher | : American Mathematical Soc. |
Total Pages | : 250 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0821808982 |
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.
Author | : Daniel S. Freed |
Publisher | : American Mathematical Society, IAS/Park City Mathematics Institute |
Total Pages | : 476 |
Release | : 2021-12-02 |
Genre | : Mathematics |
ISBN | : 1470461234 |
This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.
Author | : Heather A. Dye |
Publisher | : CRC Press |
Total Pages | : 256 |
Release | : 2018-09-03 |
Genre | : Mathematics |
ISBN | : 1315360098 |
The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra. The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.
Author | : Louis H. Kauffman |
Publisher | : World Scientific |
Total Pages | : 577 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 9814313009 |
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.