Quantum Invariants: A Study of Knots, 3-manifolds, and Their SetsWorld Scientific, 2002 - 508 páginas 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 ChernOCoSimons field theory and the WessOCoZuminoOCoWitten model are described as the physical background of the invariants. Contents: Knots and Polynomial Invariants; Braids and Representations of the Braid Groups; Operator Invariants of Tangles via Sliced Diagrams; Ribbon Hopf Algebras and Invariants of Links; Monodromy Representations of the Braid Groups Derived from the KnizhnikOCoZamolodchikov Equation; The Kontsevich Invariant; Vassiliev Invariants; Quantum Invariants of 3-Manifolds; Perturbative Invariants of Knots and 3-Manifolds; The LMO Invariant; Finite Type Invariants of Integral Homology 3-Spheres. Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics." |
Contenido
Chapter | 1 |
TGM | 5 |
Braids and representations of the braid groups | 23 |
Operator invariants of tangles via sliced diagrams | 41 |
Ribbon Hopf algebras and invariants of links | 63 |
Monodromy representations of the braid groups derived | 99 |
The Kontsevich invariant | 133 |
Vassiliev invariants | 175 |
Appendix A The quantum group U₁sl2 | 333 |
Uçsl2 at a root of unity is a ribbon Hopf algebra | 342 |
Appendix B The quantum sl3 invariant via a linear skein | 349 |
Associators | 365 |
A descending series of equivalence relations among knots | 379 |
Computing the Kontsevich and the LMO invariants of tree claspers | 399 |
Perturbative expansion | 417 |
Appendix G Computations for the perturbative invariant | 437 |
Quantum invariants of 3manifolds | 211 |
Perturbative invariants of knots | 247 |
The LMO invariant | 269 |
Chapter 11 | 305 |
Appendix H The quantum sl2 invariant and the Kauffman bracket | 457 |
484 | |
Otras ediciones - Ver todas
Quantum Invariants: A Study of Knots, 3-manifolds, and Their Sets Tomotada Ohtsuki Vista previa limitada - 2002 |
Quantum Invariants: A Study of Knots, 3-Manifolds, and Their Sets Tomotada Ohtsuki Vista previa limitada - 2001 |
Términos y frases comunes
1-tangle 3-manifold A(S¹ Alexander polynomial braid group Chern-Simons chord coefficients component computation configuration space defined definition denotes field theory Figure finite type invariants framed knot framed Kontsevich invariant framed link Further Gaussian integral Hence homeomorphic homology 3-spheres implies integral homology 3-spheres integral surgery invariant of degree invariants of 3-manifolds invariants of knots isomorphism isotopy invariant Jacobi diagrams Jones polynomial Kauffman bracket KII move KZ equation left hand side Lemma Lie algebra linear map linear sum LMO invariant matrix modulo monodromy obtained from S3 operator invariant path integral proof of Theorem Proposition quantum invariants quantum sl2 quasi-bialgebra quasi-tangle quasi-triangular quotient Reidemeister moves required formula ribbon Hopf algebra right hand side Section shown strands subspace tangle diagram topological invariant tree clasper trivalent vertices trivial knot uni-trivalent graph V₁ Vassiliev invariant vector space vertex weight system ZLMO
Pasajes populares
Página 464 - Axelrod, S., Delia Pietra, S., Witten, E., Geometric quantization of ChernSimons gauge theory, J. Differential Geom. 33 (1991) 787-902.
Página 463 - Alexander, JW, Topological invariants of knots and links Trans.
Página 466 - Drinfeld, VG Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl., 32:254-258, 1985.
Página 465 - Belavin, AA, Drinfeld, VG: Solutions of the classical Yang-Baxter equation for simple Lie algebras.
Referencias a este libro
LinKnot: Knot Theory by Computer Slavik V. Jablan,Radmila Sazdanović Sin vista previa disponible - 2007 |