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Preface xi
Part 1 Abelian Covers 1
Chapter 1 Links 3
1.1 Basic notions 3
1.2 The link group 5
1.3 Homology boundary links 10
1.4 Z/2Z-boundary links 11
1.5 Isotopy, concordance and I-equivalence 13
1.6 Link homotopy and surgery 16
1.7 Ribbon links 18
1.8 Link-symmetric groups 24
1.9 Link composition 25
Chapter 2 Homology and Duality in Covers 27
2.1 Homology and cohomology with local coefficients 27
2.2 Covers of link exteriors 28
2.3 Some terminology and notation 30
2.4 Poincaré duality and the Blanchfield pairings 30
2.5 The total linking number cover 33
2.6 The maximal abelian cover 35
2.7 Boundary 1-links 36
2.8 Concordance 38
2.9 Additivity 40
2.10 Signatures 42
Chapter 3 Determinantal Invariants 47
3.1 Elementary ideals 47
3.2 The Elementary Divisor Theorem 54
3.3 Extensions 56
3.4 Reidemeister-Franz torsion 59
3.5 Steinitz-Fox-Smythe invariants 61
3.6 1- and 2-dimensional rings 63
3.7 Bilinear pairings 66
Chapter 4 The Maximal Abelian Cover 69
4.1 Metabelian groups and the Crowell sequence 69
4.2 Free metabelian groups 71
4.3 Link module sequences 73
4.4 Localization of link module sequences 76
4.5 Chen groups 78
4.6 Applications to links 78
4.7 Chen groups, nullity and longitudes 83
4.8 I-equivalence 87
4.9 The sign-determined Alexander polynomial 89
4.10 Higher dimensional links 91
Chapter 5 Sublinks and Other Abelian Covers 95
5.1 The Torres conditions 95
5.2 Torsion again 100
5.3 Partial derivatives 103
5.4 The total linking number cover 105
5.5 Murasugi nullity 108
5.6 Fibred links 110
5.7 Finite abelian covers 113
5.8 Cyclic branched covers 119
5.9 Families of coverings 122
Chapter 6 Twisted Polynomial Invariants 125
6.1 Definition in terms of local coefficients 125
6.2 Presentations 127
6.3 Reidemeister-Franz torsion 129
6.4 Duals and pairings 130
6.5 Reciprocity 132
6.6 Applications 136
Part 2 Applications: Special Cases and Symmetries 141
Chapter 7 Knot Modules 143
7.1 Knot modules 143
7.2 A Dedekind criterion 145
7.3 Cyclic modules 147
7.4 Recovering the module from the polynomial 150
7.5 Homogeneity and realizing π-primary sequences 152
7.6 The Blanchfield pairing 154
7.7 Blanchfield pairings and Seifert matrices 159
7.8 Branched covers 161
7.9 Alexander polynomials of ribbon links 163
Chapter 8 Links with Two Components 167
8.1 Bailey's Theorem 167
8.2 Consequences of Bailey's Theorem 172
8.3 The Blanchfield pairing 176
8.4 Links with Alexander polynomial 0 178
8.5 2-Component Z/2Z-boundary links 181
8.6 Topological concordance and F - isotopy 183
8.7 Some examples 184
Chapter 9 Symmetries 189
9.1 Basic notions 189
9.2 Symmetries of knot types 190
9.3 Group actions on links 196
9.4 Strong symmetries 197
9.5 Semifree periods - the Murasugi conditions 199
9.6 Semifree periods and splitting fields 205
9.7 Links with infinitely many semifree periods 208
9.8 Knots with free periods 212
9.9 Equivariant concordance 215
Chapter 10 Singularities of Plane Algebraic Curves 219
10.1 Algebraic curves 219
10.2 Power series 222
10.3 Puiseux series 226
10.4 The Milnor number 230
10.5 The conductor 234
10.6 Resolution of singularities 239
10.7 The Gauβ-Manin connection 240
10.8 The weighted homogeneous case 242
10.9 An hermitean pairing 245
Part 3 Free Covers, Nilpotent Quotients and Completion 247
Chapter 11 Free Covers 249
11.1 Free group rings 249
11.2 Z[F(μ)]-modules 251
11.3 The Sato property 257
11.4 The Farber derivations 259
11.5 The maximal free cover and duality 260
11.6 The classical case 264
11.7 The case n = 2 266
11.8 An unlinking theorem 266
11.9 Patterns and calibrations 268
11.10 Concordance 270
Chapter 12 Nilpotent Quotients 273
12.1 Massey products 273
12.2 Products, the Dwyer filtration and groups 275
12.3 Mod-p analogues 277
12.4 The graded Lie algebra of a group 278
12.5 DGAs and minimal models 279
12.6 Free derivatives 282
12.7 Milnor invariants 283
12.8 Link homotopy and the Milnor group 288
12.9 Variants of the Milnor invariants 290
12.10 Solvable quotients and covering spaces 291
Chapter 13 Algebraic Closure 293
13.1 Homological localization 293
13.2 The nilpotent completion of a group 294
13.3 The algebraic closure of a group 295
13.4 Complements on F(μ) 301
13.5 Other notions of closure 303
13.6 Orr invariants and cS H B-links 304
Chapter 14 Disc Links 307
14.1 Disc links and string links 307
14.2 Longitudes 309
14.3 Concordance and the Artin representation 310
14.4 Homotopy 314
14.5 Milnor invariants again 315
14.6 The Gassner representation 316
14.7 High dimensions 319
Bibliography 323
Index 347
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Add Algebraic Invariants of Links, Algebraic Invariants of Links This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-th, Algebraic Invariants of Links to the inventory that you are selling on WonderClubX
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Add Algebraic Invariants of Links, Algebraic Invariants of Links This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-th, Algebraic Invariants of Links to your collection on WonderClub |