Screen LibraryAlgebraic 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-theorists, such as longitudes, the homological complexity of many-variable Laure...
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| Main Authors: | |
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| Corporate Authors: | |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; Hackensack, N.J. : |
| Publication Dates: | 2012. |
| Literature type: | eBook |
| Language: | English |
| Edition: | Second edition. |
| Series: |
K & E series on knots and everything ;
v. 52 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/8493#t=toc |
| 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. |
| Carrier Form: | 1 online resource (xiv,353pages) : illustrations. |
| Bibliography: | Includes bibliographical references (pages 323-346) and index. |
| ISBN: | 9789814407397 (electronic bk.) |
| CLC: | O189 |
| Contents: |
ch. 1. Links. 1.1. Basic notions. 1.2. The link group. 1.3. Homology boundary links. 1.4. Z/2Z-boundary links. 1.5. Isotopy, concordance and I-equivalence. 1.6. Link homotopy and surgery. 1.7. Ribbon links. 1.8. Link-symmetric groups. 1.9. Link composition -- ch. 2. Homology and duality in covers. 2.1. Homology and cohomology with local coefficient. 2.2. Covers of link exteriors. 2.3. Some terminology and notation. 2.4. Poincare duality and the Blanchfield pairings. 2.5. The total linking number cover. 2.6. The maximal abelian cover. 2.7. Boundary 1-links. 2.8. Concordance. 2.9. Additivity. 2.10. Signatures -- ch. 3. Determinantal invariants. 3.1. Elementary ideals. 3.2. The Elementary Divisor Theorem. 3.3. Extensions. 3.4. Reidemeister-Franz torsion. 3.5. Steinitz-Fox-Smythe invariants. 3.6. 1- and 2-dimensional rings. 3.7. Bilinear pairings -- ch. 4. The maximal abelian cover. 4.1. Metabelian groups and the Crowell sequence. 4.2. Free metabelian groups. 4.3. Link module sequences. 4.4. Localization of link module sequences. 4.5. Chen groups. 4.6. Applications to links. 4.7. Chen groups, nullity and longitudes. 4.8. I-equivalence. 4.9. The sign-determined Alexander polynomial. 4.10. Higher dimensional links -- ch. 5. Sublinks and other abelian covers. 5.1. The Torres conditions. 5.2. Torsion again. 5.3. Partial derivatives. 5.4. The total linking number cover. 5.5. Murasugi nullity. 5.6. Fibred links. 5.7. Finite abelian covers. 5.8. Cyclic branched covers. 5.9. Families of coverings -- ch. 6. Twisted polynomial invariants. 6.1. Definition in terms of local coefficients. 6.2. Presentations. 6.3. Reidemeister-Franz torsion. 6.4. Duals and pairings. 6.5. Reciprocity. 6.6. Applications -- ch. 7. Knot modules. 7.1. Knot modules. 7.2. A Dedekind criterion. 7.3. Cyclic modules. 7.4. Recovering the module from the polynomial. 7.5. Homogeneity and realizing [symbol]-primary sequences. 7.6. The Blanchfield pairing. 7.7. Blanchfield pairings and Seifert matrices. 7.8. Branched covers. 7.9. Alexander polynomials of ribbon links. ch. 8. Links with two components. 8.1. Bailey's Theorem. 8.2. Consequences of Bailey's Theorem. 8.3. The Blanchfield pairing. 8.4. Links with Alexander polynomial 0. 8.5. 2-component Z/2Z-boundary links. 8.6. Topological concordance and F-isotopy. 8.7. Some examples -- ch. 9. Symmetries. 9.1. Basic notions. 9.2. Symmetries of knot types. 9.3. Group actions on links. 9.4. Strong symmetries. 9.5. Semifree periods - the Murasugi conditions. 9.6. Semifree periods and splitting fields. 9.7. Links with infinitely many semifree periods. 9.8. Knots with free periods. 9.9. Equivariant concordance -- ch. 10. Singularities of plane algebraic curves. 10.1. Algebraic curves. 10.2. Power series. 10.3. Puiseux series. 10.4. The Milnor number. 10.5. The conductor. 10.6. Resolution of singularities. 10.7. The Gau[symbol]-Manin connection. 10.8. The weighted homogeneous case. 10.9. An hermitean pairing -- ch. 11. Free covers. 11.1. Free group rings. 11.2. [symbol]-modules. 11.3. The Sato property. 11.4. The Farber derivations. 11.5. The maximal free cover and duality. 11.6. The classical case. 11.7. The case n = 2. 11.8. An unlinking theorem. 11.9. Patterns and calibrations. 11.10. Concordance -- ch. 12. Nilpotent quotients. 12.1. Massey products. 12.2. Products, the Dwyer filtration and gropes. 12.3. Mod-p analogues. 12.4. The graded Lie algebra of a group. 12.5. DGAs and minimal models. 12.6. Free derivatives. 12.7. Milnor invariants. 12.8. Link homotopy and the Milnor group. 12.9. Variants of the Milnor invariants. 12.10. Solvable quotients and covering spaces -- ch. 13. Algebraic closure. 13.1. Homological localization. 13.2. The nilpotent completion of a group. 13.3. The algebraic closure of a group. 13.4. Complements on [symbol]. 13.5. Other notions of closure. 13.6. Orr invariants and cSHB-links -- ch. 14. Disc links. 14.1. Disc links and string links. 14.2. Longitudes. 14.3. Concordance and the Artin representation. 14.4. Homotopy. 14.5. Milnor invariants again. 14.6. The Gassner representation. 14.7. High dimensions. |