Quantum Invariants: A Study Of Knots, 3-manifolds, And Their Sets A Study of Knots, 3-Manifolds, and Their Sets

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. Quantum and related invariants of knots and 3-manifolds are presented, and polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants derived from representation of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. Discussion encompasses the Kontsevich invariants, the theory of Vassiliev invariants, the LMO invariants, and finite type invariants of 3-manifolds. The Chern- Simons field theory and the Wess-Zumino-Witten model are described as the physical background of the invariants. The author is affiliated with the Tokyo Institute of Technology, Japan. Annotation c. Book News, Inc., Portland, OR (booknews.com) 5.5 Relation to braid representations derived from the quantum groupChapter 6 The Kontsevich invariant; 6.1 Jacobi diagrams; 6.2 The Kontsevich invariant derived from the formal KZ equation; 6.3 Quasi-tangles and their sliced diagrams; 6.4 Combinatorial definition of the framed Kontsevich invariant; 6.5 Properties of the framed Kontsevich invariant; 6.6 Universality of the Kontsevich invariant among quantum invariants; Chapter 7 Vassiliev invariants; 7.1 Definition and fundamental properties of Vassiliev invariants; 7.2 Universality of the Kontsevich invariant among Vassiliev invariants Preface; Contents; Chapter 1 Knots and polynomial invariants; 1.1 Knots and their diagrams; 1.2 The Jones polynomial; 1.3 The Alexander polynomial; Chapter 2 Braids and representations of the braid groups; 2.1 Braids and braid groups; 2.2 Representations of the braid groups via R matrices; 2.3 Burau representation of the braid groups; Chapter 3 Operator invariants of tangles via sliced diagrams; 3.1 Tangles and their sliced diagrams; 3.2 Operator invariants of unoriented tangles; 3.3 Operator invariants of oriented tangles; Chapter 4 Ribbon Hopf algebras and invariants of links 9.3 A relation between perturbative invariants of knots and homology 3-spheresChapter 10 The LMO invariant; 10.1 Properties of the framed Kontsevich invariant; 10.2 Definition of the LMO invariant; 10.3 Universality of the LMO invariant among perturbative invariants; 10.4 Aarhus integral; Chapter 11 Finite type invariants of integral homology 3-spheres; 11.1 Definition of finite type invariants; 11.2 Universality of the LMO invariant among finite type invariants; 11.3 A descending series of equivalence relations among homology 3-spheres; Appendix A The quantum group Uq(sl2) 4.1 Ribbon Hopf algebras4.2 Invariants of links in ribbon Hopf algebras; 4.3 Operator invariants of tangles derived from ribbon Hopf algebras; 4.4 The quantum group Uq(sl2) at a generic q; 4.5 The quantum group Uc(sl2) at a root of unity C; Chapter 5 Monodromy representations of the braid groups derived from the Knizhnik-Zamolodchikov equation; 5.1 Representations of braid groups derived from the KZ equation; 5.2 Computing monodromies of the KZ equation; 5.3 Combinatorial reconstruction of the monodromy representations; 5.4 Quasi-triangular quasi-bialgebra 7.3 A descending series of equivalence relations among knots7.4 Extending the set of knots by Gauss diagrams; 7.5 Vassiliev invariants as mapping degrees on configuration spaces; Chapter 8 Quantum invariants of 3-manifolds; 8.1 3-manifolds and their surgery presentations; 8.2 The quantum SU(2) and SO(3) invariants via linear skein; 8.3 Quantum invariants of 3-manifolds via quantum invariants of links; Chapter 9 Perturbative invariants of knots and 3-manifolds; 9.1 Perturbative invariants of knots; 9.2 Perturbative invariants of homology 3-spheres In knot theory we study knots and knot types (i.e., isotopy classes of knots) as mathematical objects. A.1 Uq(sl2) at a generic q is a ribbon Hopf algebra