量子不變式

內(nèi)容概要

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.

書(shū)籍目錄

PrefaceChapter 1  Knots and polynomialinvariants　1.1  Knots and their diagrams　1.2  The Jones polynomial　1.3  The Alexander polynomialChapter 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 groupsChapter 3  Operator invariants of tangles via　3.1  Tangles and their sliced diagrams　3.2  Operator invariants of unoriented tangles  　3.3  Operator invariants of oriented tanglessliced diagramsChapter 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 Chapter 5  Monodromy representations of the braid groups    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　derived　5.5  Relation to braid representations derived from the quantum groupChapter 6  The Kontsevieh 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 invariantsChapter 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 linksChapter 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　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 integralChapter 11  Finite type invariants of integral homology 3-spher　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-spherAppendix A  The quantum group Uq（sl2）　　A.1  Uq（sl2）at a generic q is a ribbon Hopf algebra  　A.2　U（sl2）at a root of unity  is a ribbon Hopf algebra　A.3　Exceptional representations of U~（sl2）at  = -1 Appendix B  The quantum sl3 invarlant via linear skeinAppendix C  Braid representations for the Alexander polyAppendix D  AssociatorsAppendix E  ClaspersAppendix F  Physical backgroundAppendix G  Computations for the perturbative invariantAppendix H  The quantum sl2 invariant and the KauffmanBibliographyNotationIndex

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