VIRTUAL KNOTS : THE STATE OF THE ART
Vassily Olegovich. Manturov
Bok Engelsk 2012 · Electronic books.
| Annen tittel | |
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| Utgitt | Singapore : : World Scientific Publishing Company, , 2012.
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| Omfang | 1 online resource (553 p.)
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| Opplysninger | Description based upon print version of record.. - Preface by Louis H. Kauffman; Dedication; Preface by Vassily Olegovich Manturov & Denis Petrovich Ilyutko; Acknowledgments; Contents; 1. Basic Definitions and Notions; 1.1 Classical knots; 1.2 Virtual knots; 1.3 Self-linking number; 2. Virtual Knots and Three-Dimensional Topology; 2.1 Introduction; 2.2 The Kuperberg theorem; 2.3 Genus of a virtual knot; 2.3.1 Two types of connected sums; 2.3.2 The proof plan of Theorem 2.5; 2.3.3 The process of destabilization; 2.4 Recognition of virtual links; 3. Quandles (Distributive Groupoids) in Virtual Knot Theory; 3.1 Introduction. - 3.2 Quandles and their generalizations3.2.1 Geometric description of the quandle; 3.2.2 Algebraic description of the quandle; 3.2.3 The virtual quandle; 3.2.4 The coloring invariant; 3.2.5 The Alexander virtual module; 3.2.5.1 An additive approach; 3.2.5.2 Multiplicative approach; 3.3 Long virtual knots; 3.4 Virtual knots and infinite-dimensional Lie algebras; 3.4.1 Preliminaries; 3.4.2 Generalizations; 3.5 Hierarchy of virtual knots; 3.5.1 Flat virtual knots; 3.5.2 Algebraic formalism; 3.5.3 Examples; 4. The Jones-Kauffman Polynomial: Atoms; 4.1 Introduction; 4.2 Basic definitions. - 4.2.1 Virtualization and mutation4.2.2 Atoms and knots; 4.2.3 Virtual diagrams and atoms; 4.2.4 Chord diagrams; 4.2.5 Passage from atoms to chord diagrams; 4.2.6 Spanning tree for the Kauffman bracket polynomial; 4.3 The polynomial Ξ: minimality problems; 4.3.1 The leading and lowest terms of the Kauffman bracket polynomial; 4.3.2 The polynomial Ξ; 4.3.3 Examples of applications of the polynomial Ξ; 4.3.4 A surface bracket and the invariant Ξ; 4.4 Rigid virtual knots; 4.4.1 Kauffman bracket for rigid knots; 4.4.2 Minimality properties; 4.5 Minimal diagrams of long virtual knots. - 5. Khovanov Homology5.1 Introduction; 5.2 Basic constructions: The Jones polynomial; 5.3 Khovanov homology with Z2-coefficients; 5.4 Khovanov homology of double knots; 5.5 Khovanov homology and atoms; 5.6 Khovanov homology and parity; 5.7 Khovanov homology for virtual links; 5.7.1 Atoms and twisted virtual knots; 5.7.2 Khovanov complex for virtual knots; 5.8 Spanning tree for Khovanov complex; 5.9 The Khovanov polynomial and Frobenius extensions; 5.9.1 Frobenius extensions; 5.9.2 Khovanov construction for Frobenius extensions; 5.9.3 Geometrical generalizations by means of atoms. - 5.9.4 Algebraic generalizations5.10 Minimal diagrams of links; 6. Virtual Braids; 6.1 Introduction; 6.2 Definitions of virtual braids; 6.3 Virtual braids and virtual knots; 6.3.1 Closure of virtual braids; 6.3.2 Burau representation and its generalizations; 6.4 The Kauffman bracket polynomial for braids; 6.5 Invariants of virtual braids; 6.5.1 The construction of the main invariant; 6.5.2 Representation of virtual braid group; 6.5.3 On completeness in the classical case; 6.5.4 First fruits; 6.5.5 Completeness for the case of two-strand braids; 7. Vassiliev's Invariants and Framed Graphs. - 7.1 Introduction. - 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. Vir
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| Emner | |
| Sjanger | |
| Dewey | |
| ISBN | 9789814401128
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