A 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.