Title Introduction to Vassiliev knot invariants / S. Chmutov, S. Duzhin, J. Mostovoy

Published New York : Cambridge University Press, 2012

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Description 1 online resource Contents Cover; INTRODUCTION TO VASSILIEV KNOT INVARIANTS; Title; Copyright; Dedication; Contents; Preface; 1 Knots and their relatives; 1.1 Definitions and examples; 1.2 Plane knot diagrams; 1.3 Inverses and mirror images; 1.4 Knot tables; 1.5 Algebra of knots; 1.6 Tangles, string links and braids; 1.7 Variations; Exercises; 2 Knot invariants; 2.1 Definition and first examples; 2.2 Linking number; 2.3 The Conway polynomial; 2.4 The Jones polynomial; 2.5 Algebra of knot invariants; 2.6 Quantum invariants; 2.7 Two-variable link polynomials; Exercises; 3 Finite type invariants 3.1 Definition of Vassiliev invariants3.2 Algebra of Vassiliev invariants; 3.3 Vassiliev invariants of degrees 0, 1 and 2; 3.4 Chord diagrams; 3.5 Invariants of framed knots; 3.6 Classical knot polynomials as Vassiliev invariants; 3.7 Actuality tables; 3.8 Vassiliev invariants of tangles; Exercises; 4 Chord diagrams; 4.1 Four- and one-term relations; 4.2 The Fundamental Theorem; 4.3 Bialgebras of knots and of Vassiliev knot invariants; 4.4 Bialgebra of chord diagrams; 4.5 Bialgebra of weight systems; 4.6 Primitive elements in A; 4.7 Linear chord diagrams; 4.8 Intersection graphs; Exercises 5 Jacobi diagrams5.1 Closed Jacobi diagrams; 5.2 IHX and AS relations; 5.3 Isomorphism A?C; 5.4 Product and coproduct in C; 5.5 Primitive subspace of C; 5.6 Open Jacobi diagrams; 5.7 Linear isomorphism B?C; 5.8 More on the relation between B and C; 5.9 The three algebras in small degrees; 5.10 Jacobi diagrams for tangles; 5.11 Horizontal chord diagrams; Exercises; 6 Lie algebra weight systems; 6.1 Lie algebra weight systems for the algebra A; 6.2 Lie algebra weight systems for the algebra C; 6.3 Lie algebra weight systems for the algebra B; 6.4 Lie superalgebra weight systems; Exercises 7 Algebra of 3-graphs7.1 The space of 3-graphs; 7.2 Edge multiplication; 7.3 Vertex multiplication; 7.4 Action of? on the primitive space P; 7.5 Lie algebra weight systems for the algebra?; 7.6 Vogel's algebra?; Exercises; 8 The Kontsevich integral; 8.1 First examples; 8.2 The construction; 8.3 Example of calculation; 8.4 The Kontsevich integral for tangles; 8.5 Convergence of the integral; 8.6 Invariance of the integral; 8.7 Changing the number of critical points; 8.8 The universal Vassiliev invariant; 8.9 Symmetries and the group-like property of Z(K) 8.10 Towards the combinatorial Kontsevich integralExercises; 9 Framed knots and cabling operations; 9.1 Framed version of the Kontsevich integral; 9.2 Cabling operations; 9.3 Cabling operations and the Kontsevich integral; 9.4 Cablings of the Lie algebra weight systems; Exercises; 10 The Drinfeld associator; 10.1 The KZ equation and iterated integrals; 10.2 Calculation of the KZ Drinfeld associator; 10.3 Combinatorial construction of the Kontsevich integral; 10.4 General associators; Exercises; 11 The Kontsevich integral: advanced features; 11.1 Mutation; 11.2 Canonical Vassiliev invariants Summary "With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced. This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots"-- Provided by publisher Bibliography Includes bibliographical references and index Notes Print version record Subject Knot theory. Invariants. MATHEMATICS -- Topology. Invariantes Nudos, Teoría de Invariants Knot theory Knotentheorie Knopentheorie. Form Electronic book Author Duzhin, S. V. (Sergeĭ Vasilʹevich), 1956- Mostovoy, J. (Jacob) ISBN 9781139424097 1139424092 9781139107846 1139107844 9781139422055 1139422057 1139420003 9781139420006 1280685190 9781280685194

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