Author | Kawauchi, Akio. author |
---|---|
Title | A Survey of Knot Theory [electronic resource] / by Akio Kawauchi |
Imprint | Basel : Birkhรคuser Basel, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9227-8 |
Descript | XXI, 423 p. online resource |
0 Fundamentals of knot theory -- 0.1 Spaces -- 0.2 Manifolds and submanifolds -- 0.3 Knots and links -- Supplementary notes for Chapter 0 -- 1 Presentations -- 1.1 Regular presentations -- 1.2 Braid presentations -- 1.3 Bridge presentations -- Supplementary notes for Chapter 1 -- 2 Standard examples -- 2.1 Two-bridge links -- 2.2 Torus links -- 2.3 Pretzel links -- Supplementary notes for Chapter 2 -- 3 Compositions and decompositions -- 3.1 Compositions of links -- 3.2 Decompositions of links -- 3.3 Definition of a tangle and examples -- 3.4 How to judge the non-splittability of a link -- 3.5 How to judge the primeness of a link -- 3.6 How to judge the hyperbolicity of a link -- 3.7 Non-triviality of a link -- 3.8 Conway mutation -- Supplementary notes for Chapter 3 -- 4 Seifert surfaces I: a topological approach -- 4.1 Definition and existence of Seifert surfaces -- 4.2 The Murasugi sum -- 4.3 Sutured manifolds -- Supplementary notes for Chapter 4 -- 5 Seifert surfaces II: an algebraic approach -- 5.1 The Seifert matrix -- 5.2 S-equivalence -- 5.3 Number-theoretic invariants -- 5.4 The reduced link module -- 5.5 The homology of a branched cyclic covering manifold -- Supplementary notes for Chapter 5 -- 6 The fundamental group -- 6.1 Link groups and link group systems -- 6.2 Presentations of a link group -- 6.3 Subgroups and quotient groups of a link group -- Supplementary notes for Chapter 6 -- 7 Multi-variable Alexander polynomials -- 7.1 The Alexander module -- 7.2 Invariants of a A-module -- 7.3 Graded Alexander polynomials -- 7.4 Torres conditions -- Supplementary notes for Chapter 7 -- 8 Jones type polynomials I: a topological approach -- 8.1 The Jones polynomial -- 8.2 The skein polynomial -- 8.3 The Q and Kauffman polynomials -- 8.4 Properties of the polynomial invariants -- 8.5 The skein polynomial via a state model -- Supplementary notes for Chapter 8 -- 9 Jones type polynomials II: an algebraic approach -- 9.1 Preliminaries from representation theory -- 9.2 Link invariants of trace type -- 9.3 The skein polynomial as a link invariant of trace type -- 9.4 The Temperley-Lieb algebra -- Supplementary notes for Chapter 9 -- 10 Symmetries -- 10.1 Periodic knots -- 10.2 Freely periodic knots -- 10.3 Invertible knots -- 10.4 Amphicheiral knots -- 10.5 Symmetries of a hyperbolic knot -- 10.6 The symmetry group -- 10.7 Canonical decompositions and symmetry -- Supplementary notes for Chapter 10 -- 11 Local transformations -- 11.1 Unknotting operations -- 11.2 Properties of X-Gordian distance -- 11.3 Properties of ?-Gordian distance -- 11.4 Properties of #-Gordian distance -- 11.5 Estimation of the X-unknotting number -- 11.6 Local transformations of links -- Supplementary notes for Chapter 11 -- 12 Cobordisms -- 12.1 The knot cobordism group -- 12.2 The matrix cobordism group -- 12.3 Link cobordism -- Supplementary notes for Chapter 12 -- 13 Two-knots I: a topological approach -- 13.1 A normal form -- 13.2 Constructing 2-knots -- 13.3 Seifert hypersurfaces -- 13.4 Exteriors of 2-knots -- 13.5 Cyclic covering spaces -- 13.6 The k-invariant -- 13.7 Ribbon presentations -- Supplementary notes for Chapter 13 -- 14 Two-knots II: an algebraic approach -- 14.1 High-dimensional knot groups -- 14.2 Ribbon 2-knot groups -- 14.3 Torsion elements and the deficiency of 2-knot groups -- Supplementary notes for Chapter 14 -- 15 Knot theory of spatial graphs -- 15.1 Topology of molecules -- 15.2 Uses of the notion of equivalence -- 15.3 Uses of the notion of neighborhood-equivalence -- Supplementary notes for Chapter 15 -- 16 Vassiliev-Gusarov invariants -- 16.1 Vassiliev-Gusarov algebra -- 16.2 Vassiliev-Gusarov invariants and Jones type polynomials -- 16.3 Kontsevichโs iterated integral invariant -- 16.4 Numerical invariants not of Vassiliev-Gusarov type -- Supplementary notes for Chapter 16 -- Appendix A The equivalence of several notions of โlink equivalenceโ -- Appendix B Covering spaces -- Appendix C Canonical decompositions of 3-manifolds -- Appendix D Heegaard splittings and Dehn surgery descriptions -- Appendix E The Blanchfield duality theorem -- Appendix F Tables of data -- References