Algebraic Invariants of Links

by Hillman, Jonathan

• ISBN13: 9789814407380
• ISBN10: 9814407380
• Edition: 2nd
• Format: Hardcover
• Copyright: 2012-06-15
• Publisher: World Scientific Pub Co Inc

Summary

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-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology.This second edition introduces two new chapters - twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

Table of Contents

Preface	p. xi
Abelian Covers	p. 1
Links	p. 3
Basic notions	p. 3
The link group	p. 5
Homology boundary links	p. 10
Z/2Z-boundary links	p. 11
Isotopy, concordance and I-equivalence	p. 13
Link homotopy and surgery	p. 16
Ribbon links	p. 18
Link-symmetric groups	p. 24
Link composition	p. 25
Homology and Duality in Covers	p. 27
Homology and cohomology with local coefficients	p. 27
Covers of link exteriors	p. 28
Some terminology and notation	p. 30
Poincaré duality and the Blanchfield pairings	p. 30
The total linking number cover	p. 33
The maximal abelian cover	p. 35
Boundary 1-links	p. 36
Concordance	p. 38
Additivity	p. 40
Signatures	p. 42
Determinantal Invariants	p. 47
Elementary ideals	p. 47
The Elementary Divisor Theorem	p. 54
Extensions	p. 56
Reidemeister-Franz torsion	p. 59
Steinitz-Fox-Smythe invariants	p. 61
1- and 2-dimensional rings	p. 63
Bilinear pairings	p. 66
The Maximal Abelian Cover	p. 69
Metabelian groups and the Crowell sequence	p. 69
Free metabelian groups	p. 71
Link module sequences	p. 73
Localization of link module sequences	p. 76
Chen groups	p. 78
Applications to links	p. 78
Chen groups, nullity and longitudes	p. 83
I-equivalence	p. 87
The sign-determined Alexander polynomial	p. 89
Higher dimensional links	p. 91
Sublinks and Other Abelian Covers	p. 95
The Torres conditions	p. 95
Torsion again	p. 100
Partial derivatives	p. 103
The total linking number cover	p. 105
Murasugi nullity	p. 108
Fibred links	p. 110
Finite abelian covers	p. 113
Cyclic branched covers	p. 119
Families of coverings	p. 122
Twisted Polynomial Invariants	p. 125
Definition in terms of local coefficients	p. 125
Presentations	p. 127
Reidemeister-Franz torsion	p. 129
Duals and pairings	p. 130
Reciprocity	p. 132
Applications	p. 136
Applications: Special Cases and Symmetries	p. 141
Knot Modules	p. 143
Knot modules	p. 143
A Dedekind criterion	p. 145
Cyclic modules	p. 147
Recovering the module from the polynomial	p. 150
Homogeneity and realizing ¿-primary sequences	p. 152
The Blanchfield pairing	p. 154
Blanchfield pairings and Seifert matrices	p. 159
Branched covers	p. 161
Alexander polynomials of ribbon links	p. 163
Links with Two Components	p. 167
Bailey's Theorem	p. 167
Consequences of Bailey's Theorem	p. 172
The Blanchfield pairing	p. 176
Links with Alexander polynomial 0	p. 178
2-Component Z/2Z-boundary links	p. 181
Topological concordance and F - isotopy	p. 183
Some examples	p. 184
Symmetries	p. 189
Basic notions	p. 189
Symmetries of knot types	p. 190
Group actions on links	p. 196
Strong symmetries	p. 197
Semifree periods - the Murasugi conditions	p. 199
Semifree periods and splitting fields	p. 205
Links with infinitely many semifree periods	p. 208
Knots with free periods	p. 212
Equivariant concordance	p. 215
Singularities of Plane Algebraic Curves	p. 219
Algebraic curves	p. 219
Power series	p. 222
Puiseux series	p. 226
The Milnor number	p. 230
The conductor	p. 234
Resolution of singularities	p. 239
The Gauß-Manin connection	p. 240
The weighted homogeneous case	p. 242
An hermitean pairing	p. 245
Free Covers, Nilpotent Quotients and Completion	p. 247
Free Covers	p. 249
Free group rings	p. 249
Z[F(¿)]-modules	p. 251
The Sato property	p. 257
The Farber derivations	p. 259
The maximal free cover and duality	p. 260
The classical case	p. 264
The case n = 2	p. 266
An unlinking theorem	p. 266
Patterns and calibrations	p. 268
Concordance	p. 270
Nilpotent Quotients	p. 273
Massey products	p. 273
Products, the Dwyer filtration and groups	p. 275
Mod-p analogues	p. 277
The graded Lie algebra of a group	p. 278
DGAs and minimal models	p. 279
Free derivatives	p. 282
Milnor invariants	p. 283
Link homotopy and the Milnor group	p. 288
Variants of the Milnor invariants	p. 290
Solvable quotients and covering spaces	p. 291
Algebraic Closure	p. 293
Homological localization	p. 293
The nilpotent completion of a group	p. 294
The algebraic closure of a group	p. 295
Complements on F(¿)	p. 301
Other notions of closure	p. 303
Orr invariants and cS H B-links	p. 304
Disc Links	p. 307
Disc links and string links	p. 307
Longitudes	p. 309
Concordance and the Artin representation	p. 310
Homotopy	p. 314
Milnor invariants again	p. 315
The Gassner representation	p. 316
High dimensions	p. 319
Bibliography	p. 323
Index	p. 347
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