Virtual Knots

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Virtual Knots

Vasilii Olegovich Manturov,Denis Petrovich Ilyutko
Virtual Knots Book PDF

Author : Vasilii Olegovich Manturov,Denis Petrovich Ilyutko
Publisher : World Scientific
Page : 553 pages
File Size : 51,5 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.

Knots, Low-Dimensional Topology and Applications

Colin C. Adams,Cameron McA. Gordon,Vaughan F.R. Jones,Louis H. Kauffman,Sofia Lambropoulou,Kenneth C. Millett,Jozef H. Przytycki,Renzo Ricca,Radmila Sazdanovic
Knots Low Dimensional Topology and Applications Book PDF

Author : Colin C. Adams,Cameron McA. Gordon,Vaughan F.R. Jones,Louis H. Kauffman,Sofia Lambropoulou,Kenneth C. Millett,Jozef H. Przytycki,Renzo Ricca,Radmila Sazdanovic
Publisher : Springer
Page : 476 pages
File Size : 47,8 Mb
Release : 2019-06-26
Category : Mathematics
ISBN : 9783030160319

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Knots, Low-Dimensional Topology and Applications by Colin C. Adams,Cameron McA. Gordon,Vaughan F.R. Jones,Louis H. Kauffman,Sofia Lambropoulou,Kenneth C. Millett,Jozef H. Przytycki,Renzo Ricca,Radmila Sazdanovic Pdf

This proceedings volume presents a diverse collection of high-quality, state-of-the-art research and survey articles written by top experts in low-dimensional topology and its applications. The focal topics include the wide range of historical and contemporary invariants of knots and links and related topics such as three- and four-dimensional manifolds, braids, virtual knot theory, quantum invariants, braids, skein modules and knot algebras, link homology, quandles and their homology; hyperbolic knots and geometric structures of three-dimensional manifolds; the mechanism of topological surgery in physical processes, knots in Nature in the sense of physical knots with applications to polymers, DNA enzyme mechanisms, and protein structure and function. The contents is based on contributions presented at the International Conference on Knots, Low-Dimensional Topology and Applications – Knots in Hellas 2016, which was held at the International Olympic Academy in Greece in July 2016. The goal of the international conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and low-dimensional topology. This book will benefit all researchers who wish to take their research in new directions, to learn about new tools and methods, and to discover relevant and recent literature for future study.

Knots and Physics

Louis H. Kauffman
Knots and Physics Book PDF

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 865 pages
File Size : 44,8 Mb
Release : 2013
Category : Mathematics
ISBN : 9789814383004

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Knots and Physics by Louis H. Kauffman Pdf

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 an extraordinary range of interrelated topics in topology and mathematical physics.

Introductory Lectures on Knot Theory

Louis H. Kauffman
Introductory Lectures on Knot Theory Book PDF

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 578 pages
File Size : 42,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814307994

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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.

Encyclopedia of Knot Theory

Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson
Encyclopedia of Knot Theory Book PDF

Author : Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson
Publisher : CRC Press
Page : 954 pages
File Size : 41,5 Mb
Release : 2021-02-10
Category : Education
ISBN : 9781000222388

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Encyclopedia of Knot Theory by Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson Pdf

"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject." – Ed Witten, Recipient of the Fields Medal "I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field." – Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deep mathematical connections, and critical applications. Many of the articles in this book are accessible to undergraduates who are working on research or taking an advanced undergraduate course in knot theory. More advanced articles will be useful to graduate students working on a related thesis topic, to researchers in another area of topology who are interested in current results in knot theory, and to scientists who study the topology and geometry of biopolymers. Features Provides material that is useful and accessible to undergraduates, postgraduates, and full-time researchers Topics discussed provide an excellent catalyst for students to explore meaningful research and gain confidence and commitment to pursuing advanced degrees Edited and contributed by top researchers in the field of knot theory

Handbook of Knot Theory

William Menasco,Morwen Thistlethwaite
Handbook of Knot Theory Book PDF

Author : William Menasco,Morwen Thistlethwaite
Publisher : Elsevier
Page : 502 pages
File Size : 44,5 Mb
Release : 2005-08-02
Category : Mathematics
ISBN : 0080459544

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Handbook of Knot Theory by William Menasco,Morwen Thistlethwaite Pdf

This book is a survey of current topics in the mathematical theory of knots. For a mathematician, a knot is a closed loop in 3-dimensional space: imagine knotting an extension cord and then closing it up by inserting its plug into its outlet. Knot theory is of central importance in pure and applied mathematics, as it stands at a crossroads of topology, combinatorics, algebra, mathematical physics and biochemistry. * Survey of mathematical knot theory * Articles by leading world authorities * Clear exposition, not over-technical * Accessible to readers with undergraduate background in mathematics

The Mathematics of Knots

Markus Banagl,Denis Vogel
The Mathematics of Knots Book PDF

Author : Markus Banagl,Denis Vogel
Publisher : Springer Science & Business Media
Page : 363 pages
File Size : 47,8 Mb
Release : 2010-11-25
Category : Mathematics
ISBN : 9783642156373

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The Mathematics of Knots by Markus Banagl,Denis Vogel Pdf

The present volume grew out of the Heidelberg Knot Theory Semester, organized by the editors in winter 2008/09 at Heidelberg University. The contributed papers bring the reader up to date on the currently most actively pursued areas of mathematical knot theory and its applications in mathematical physics and cell biology. Both original research and survey articles are presented; numerous illustrations support the text. The book will be of great interest to researchers in topology, geometry, and mathematical physics, graduate students specializing in knot theory, and cell biologists interested in the topology of DNA strands.

Knots, Links, Spatial Graphs, and Algebraic Invariants

Erica Flapan,Allison Henrich,Aaron Kaestner,Sam Nelson:
Knots Links Spatial Graphs and Algebraic Invariants Book PDF

Author : Erica Flapan,Allison Henrich,Aaron Kaestner,Sam Nelson:
Publisher : American Mathematical Soc.
Page : 189 pages
File Size : 44,5 Mb
Release : 2017-05-19
Category : Combinatorics -- Graph theory -- Planar graphs
ISBN : 9781470428471

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Knots, Links, Spatial Graphs, and Algebraic Invariants by Erica Flapan,Allison Henrich,Aaron Kaestner,Sam Nelson: Pdf

This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24–25, 2015, at California State University, Fullerton, CA. Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves. The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in and other 3-manifolds.

Knot Theory

Vassily Olegovich Manturov
Knot Theory Book PDF

Author : Vassily Olegovich Manturov
Publisher : CRC Press
Page : 560 pages
File Size : 40,7 Mb
Release : 2018-04-17
Category : Mathematics
ISBN : 9781351359139

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Knot Theory by Vassily Olegovich Manturov Pdf

Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and alge-bra. It suffices to mention the great progress in knot homology theory (Khovanov homology and Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give rise to strong invariants of knots and 3-manifolds, in particular, many new unknot detectors. New to this Edition is a discussion of Heegaard-Floer homology theory and A-polynomial of classical links, as well as updates throughout the text. Knot Theory, Second Edition is notable not only for its expert presentation of knot theory’s state of the art but also for its accessibility. It is valuable as a profes-sional reference and will serve equally well as a text for a course on knot theory.

Knots in Hellas '98

C. McA. Gordon
Knots in Hellas 98 Book PDF

Author : C. McA. Gordon
Publisher : World Scientific
Page : 580 pages
File Size : 50,5 Mb
Release : 2000
Category : Mathematics
ISBN : 9789810243401

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Knots in Hellas '98 by C. McA. Gordon Pdf

There have been exciting developments in the area of knot theory in recent years. They include Thurston's work on geometric structures on 3-manifolds (e.g. knot complements), Gordon?Luecke work on surgeries on knots, Jones' work on invariants of links in S3, and advances in the theory of invariants of 3-manifolds based on Jones- and Vassiliev-type invariants of links. Jones ideas and Thurston's idea are connected by the following path: hyperbolic structures, PSL(2, C) representations, character varieties, quantization of the coordinate ring of the variety to skein modules (i.e. Kauffman, bracket skein module), and finally quantum invariants of 3-manifolds. This proceedings volume covers all those exciting topics.

Virtual Knots: The State Of The Art

Manturov Vassily Olegovich,Ilyutko Denis Petrovich
Virtual Knots The State Of The Art Book PDF

Author : Manturov Vassily Olegovich,Ilyutko Denis Petrovich
Publisher : World Scientific
Page : 553 pages
File Size : 46,5 Mb
Release : 2012-09-21
Category : Mathematics
ISBN : 9789814401142

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Virtual Knots: The State Of The Art by Manturov Vassily Olegovich,Ilyutko Denis Petrovich 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 “diagramless knot theory”: such “links” 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.

Knots, Links and Their Invariants

A. B. Sossinsky
Knots Links and Their Invariants Book PDF

Author : A. B. Sossinsky
Publisher : American Mathematical Society
Page : 149 pages
File Size : 53,7 Mb
Release : 2023-05-22
Category : Mathematics
ISBN : 9781470471514

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Knots, Links and Their Invariants by A. B. Sossinsky Pdf

This book is an elementary introduction to knot theory. Unlike many other books on knot theory, this book has practically no prerequisites; it requires only basic plane and spatial Euclidean geometry but no knowledge of topology or group theory. It contains the first elementary proof of the existence of the Alexander polynomial of a knot or a link based on the Conway axioms, particularly the Conway skein relation. The book also contains an elementary exposition of the Jones polynomial, HOMFLY polynomial and Vassiliev knot invariants constructed using the Kontsevich integral. Additionally, there is a lecture introducing the braid group and shows its connection with knots and links. Other important features of the book are the large number of original illustrations, numerous exercises and the absence of any references in the first eleven lectures. The last two lectures differ from the first eleven: they comprise a sketch of non-elementary topics and a brief history of the subject, including many references.

An Interactive Introduction to Knot Theory

Inga Johnson,Allison Henrich
An Interactive Introduction to Knot Theory Book PDF

Author : Inga Johnson,Allison Henrich
Publisher : Courier Dover Publications
Page : 193 pages
File Size : 44,9 Mb
Release : 2017-01-18
Category : Mathematics
ISBN : 9780486804637

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An Interactive Introduction to Knot Theory by Inga Johnson,Allison Henrich Pdf

This well-written and engaging volume, intended for undergraduates, introduces knot theory, an area of growing interest in contemporary mathematics. The hands-on approach features many exercises to be completed by readers. Prerequisites are only a basic familiarity with linear algebra and a willingness to explore the subject in a hands-on manner. The opening chapter offers activities that explore the world of knots and links — including games with knots — and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.

An Invitation to Knot Theory

Heather A. Dye
An Invitation to Knot Theory Book PDF

Author : Heather A. Dye
Publisher : CRC Press
Page : 256 pages
File Size : 40,9 Mb
Release : 2018-09-03
Category : Mathematics
ISBN : 9781315360096

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An Invitation to Knot Theory by Heather A. Dye Pdf

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.

Surfaces in 4-Space

Scott Carter,Seiichi Kamada,Masahico Saito
Surfaces in 4 Space Book PDF

Author : Scott Carter,Seiichi Kamada,Masahico Saito
Publisher : Springer Science & Business Media
Page : 220 pages
File Size : 55,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662101629

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Surfaces in 4-Space by Scott Carter,Seiichi Kamada,Masahico Saito Pdf

Surfaces in 4-Space, written by leading specialists in the field, discusses knotted surfaces in 4-dimensional space and surveys many of the known results in the area. Results on knotted surface diagrams, constructions of knotted surfaces, classically defined invariants, and new invariants defined via quandle homology theory are presented. The last chapter comprises many recent results, and techniques for computation are presented. New tables of quandles with a few elements and the homology groups thereof are included. This book contains many new illustrations of knotted surface diagrams. The reader of the book will become intimately aware of the subtleties in going from the classical case of knotted circles in 3-space to this higher dimensional case. As a survey, the book is a guide book to the extensive literature on knotted surfaces and will become a useful reference for graduate students and researchers in mathematics and physics.