Contents -- Preface -- 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 -- 4.1 Ribbon Hopf algebras -- 4.2 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 -- 5.5 Relation to braid representations derived from the quantum group -- Chapter 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 -- 7.3 A descending series of equivalence relations among knots -- 7.4 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 -- 9.3 A relation between perturbative invariants of knots and homology 3-spheres -- Chapter 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) -- A.1 Uq(sl2) at a generic q is a ribbon Hopf algebra -- A.2 Uc(sl2) at a root of unity c is a ribbon Hopf algebra -- A.3 Exceptional representations of Uc(sl2) at c = -1 -- Appendix B The quantum sl3 invariant via a linear skein -- B.1 The quantum (sl3 V) invariant of framed links -- B.2 The quantum SU(3) and PSU(3) invariants of 3-manifolds -- Appendix C Braid representations for the Alexander polynomial -- C 1 Relation between two braid representations -- Appendix D Associators -- D.1 Drinfel'd series -- D.2 The Drinfel'd associator -- Appendix E Claspers.

E.1 Basic properties of claspers -- E.2 A descending series of equivalence relations among knots -- E.3 A descending series of equivalence relations among homology 3-spheres -- E.4 Computing the Kontsevich and the LMO invariants of tree claspers -- Appendix F Physical background -- F.1 Chern-Simons field theory -- F.2 Topological quantum field theory -- F.3 Perturbative expansion -- F.4 Conformal field theory by Wess-Zumino-Witten model -- Appendix G Computations for the perturbative invariant -- G.1 Gaussian sum -- G.2 The center of the quantum group Uq(sl2) -- G.3 The quantum (sl2 -- ak) invariant is divisible by (q - 1 )2k -- G.4 Computation of formal Gaussian integrals -- Appendix H The quantum sl2 invariant and the Kauffman bracket -- H.1 The quantum (sl2 V) invariant by the Kauffman bracket -- H.2 The quantum (sl2 Vn) invariant by the linear skein -- Bibliography -- Notation -- Index.