Vassily Manturov's Books

Knot Theory

Second Edition

By: Vassily Olegovich Manturov

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Knot Theory

Second Edition

By: Vassily Olegovich Manturov

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.

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Page Count

560

Publisher

CRC Press

Published Date

ISBN 10

1351359134

ISBN 13

9781351359139

Knot Theory

By: Vassily Olegovich Manturov,Vassily Manturov

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Knot Theory

By: Vassily Olegovich Manturov,Vassily Manturov

Since discovery of the Jones polynomial, knot theory has enjoyed a virtual explosion of important results and now plays a significant role in modern mathematics. In a unique presentation with contents not found in any other monograph, Knot Theory describes, with full proofs, the main concepts and the latest investigations in the field. The book is divided into six thematic sections. The first part discusses "pre-Vassiliev" knot theory, from knot arithmetics through the Jones polynomial and the famous Kauffman-Murasugi theorem. The second part explores braid theory, including braids in different spaces and simple word recognition algorithms. A section devoted to the Vassiliev knot invariants follows, wherein the author proves that Vassiliev invariants are stronger than all polynomial invariants and introduces Bar-Natan's theory on Lie algebra respresentations and knots. The fourth part describes a new way, proposed by the author, to encode knots by d-diagrams. This method allows the encoding of topological objects by words in a finite alphabet. Part Five delves into virtual knot theory and virtualizations of knot and link invariants. This section includes the author's own important results regarding new invariants of virtual knots. The book concludes with an introduction to knots in 3-manifolds and Legendrian knots and links, including Chekanov's differential graded algebra (DGA) construction. Knot Theory 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 professional reference and will serve equally well as a text for a course on knot theory.

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Page Count

417

Publisher

CRC Press

Published Date

ISBN 10

0203402847

ISBN 13

9780203402849

Knot Theory

By: Vassily Olegovich Manturov,Vassily Manturov

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Knot Theory

By: Vassily Olegovich Manturov,Vassily Manturov

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417

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CRC Press

Published Date

ISBN 10

0203402847

ISBN 13

9780203402849

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

By: Vassily Olegovich Manturov,Denis Fedoseev,Seongjeong Kim,Igor Nikonov

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Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

By: Vassily Olegovich Manturov,Denis Fedoseev,Seongjeong Kim,Igor Nikonov

This book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of Gnk groups, picture-valued invariants, and braids for arbitrary manifolds. Equivalence relations arising in low-dimensional topology and combinatorial group theory inevitably lead to the study of invariants, and good invariants should be strong and apparent. An interesting case of such invariants is picture-valued invariants, whose values are not algebraic objects, but geometrical constructions, like graphs or polyhedra.In 2015, V O Manturov defined a two-parametric family of groups Gnk and formulated the following principle: if dynamical systems describing a motion of n particles possess a nice codimension 1 property governed by exactly k particles then these dynamical systems possess topological invariants valued in Gnk.The book is devoted to various realisations and generalisations of this principle in the broad sense. The groups Gnk have many epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare. However, this construction does not work when we try to deal with points on a 2-surface, since there may be infinitely many geodesics passing through two points. That leads to the notion of another family of groups — Γnk, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds.

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Page Count

387

Publisher

World Scientific

Published Date

ISBN 10

9811220131

ISBN 13

9789811220135

Virtual Knots: The State Of The Art

By: Manturov Vassily Olegovich,Ilyutko Denis Petrovich

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Virtual Knots: The State Of The Art

By: Manturov Vassily Olegovich,Ilyutko Denis Petrovich

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.

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Page Count

553

Publisher

World Scientific

Published Date

ISBN 10

9814401145

ISBN 13

9789814401142

Virtual Knots: The State Of The Art

By: Manturov Vassily Olegovich,Ilyutko Denis Petrovich

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Virtual Knots: The State Of The Art

By: Manturov Vassily Olegovich,Ilyutko Denis Petrovich

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Page Count

553

Publisher

World Scientific

Published Date

ISBN 10

9814401145

ISBN 13

9789814401142

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

By: Vassily Olegovich Manturov,Denis Fedoseev,Seongjeong Kim,Igor Nikonov

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Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

By: Vassily Olegovich Manturov,Denis Fedoseev,Seongjeong Kim,Igor Nikonov

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Solve jigsaw puzzle

Age

Page Count

387

Publisher

World Scientific

Published Date

ISBN 10

9811220131

ISBN 13

9789811220135

Book's by Vassily Manturov

Knot Theory

Knot Theory

Knot Theory

Knot Theory

Knot Theory

Knot Theory

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

Virtual Knots: The State Of The Art

Virtual Knots: The State Of The Art

Virtual Knots: The State Of The Art

Virtual Knots: The State Of The Art

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

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