Survey of Knot Theory

• 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 typepolynomials.- 16.3 Kontsevich’s 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 “link equivalence”.- 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.

"The book...is in fact a small but feisty encyclopedia of the subject. It achieves its aim in a compact space by accurate statements of theorems and examples and by eschewing the details of the proofs of many theorems... A useful reference book that can serve as an introduction to many topics in the modern theory of knots." --Bulletin of the AMS