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Algebraic Invariants of Links Book

Algebraic Invariants of Links
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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
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  • Algebraic Invariants of Links
  • Written by author Jonathan Hillman
  • Published by World Scientific Publishing Company, Incorporated, 7/28/2012
  • 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
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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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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

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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

Algebraic Invariants of Links

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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

Algebraic Invariants of Links

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