Handbook of knot theory
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| Sprache: | English |
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2005
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| 245 | 1 | 0 | |a Handbook of knot theory |c ed. William Menasco ... |
| 250 | |a 1. impr. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2005 | |
| 300 | |a IX, 492 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 4 | |a Knot theory | |
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Datensatz im Suchindex
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|---|---|
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| DE-BY-UBM_katkey | 4135775 |
| DE-BY-UBM_media_number | 41612661740012 |
| _version_ | 1823054361139347456 |
| adam_text | Contents
Préface v
List of Contributors vii
1. Hyperbolic Knots 1
C. Adams
2. Braids: A Survey 19
J.S. Birman and T.E. Brendle
3. Legendrian and Transversal Knots 105
J.B. Etnyre
4. Knot Spinning 187
G. Friedman
5. The Enumeration and Classification of Knots and Links 209
J. Hoste
6. Knot Diagrammatics 233
L.H. Kauffman
7. A Survey of Classical Knot Concordance 319
C. Livingston
8. Knot Theory of Complex Plane Curves 349
L. Rudolph
9. Thin Position in the Theory of Classical Knots 429
M. Scharlemann
10. Computation of Hyperbolic Structures in Knot Theory 461
J. Weeks
Author Index 481
Subject Index 483
ix
CHAPTER 1
Hyperbolic Knots
Colin Adams
Bronfinan Science Center, Department of Mathematics, Williams Collège,
Williamstown, MA 01267, USA
E mail: colin, adams @ Williams, edu
Contents
1. Introduction 3
2. What knot and link compléments are known to be hyperbolic? 3
3. Volumes of knots 8
4. Cusps 10
5. Meridians and other cusp invariants 11
6. Geodesics and totally géodésie surfaces 13
Acknowledgements 16
Références 16
CHAPTER 2
Braids: A Survey
Joan S. Birman*
Department of Malhematics, Bamard Collège, Columbia University,
2990 Broadway, New York, NY 10027, USA
E mail: jb@math.columbia.edu
Tara E. Brendle+
Department of Mathematics, Louisiana State University, Bâton Rouge, IA 70803 4918, USA
E mail: brendle@math.lsu.edu
Contents 2i
1. Introduction 21
1.1. B , and P,, via configuration spaces 22
1.2. B« and P,, via generators and relations 24
1.3. B», and P,, as mapping class groups ;
1.4. Some examples where braiding appears in mathematics, unexpectedly ;;;;; .; 29
2. From knots to braids 29
2.1. Closed braids 30
2.2. Alexander s Theorem 35
2.3. Markov s Theorem 44
3. Braid foliations , . A 44
3.1. The Markov Theorem Without Stabilization (spécial case: the unknot) ^
3.2. The Markov Theorem Without Stabilization, gênerai case . . .[.... 56
3.3. Braids and contact structures 62
4. Représentations of the braid groups 62
4.1. A brief look at représentations of Sn • • 63
4.2. The Burau représentation and polynomial invariants ot knots ¦••¦;• • ¦ _ _ • •
4.3. Hecke algebras représentations of braid groups and polynom.al mvanants of knots 64
4.4. A topological interprétation of the Burau représentation /.. . .... 70
4.5. The Lawrence Krammer représentation
^Thefirs^uthor acknowiedges partia! support from the U.S. National Science Foundaùon under grant number
^second author was partially supported by a VIGRE postdoc under NSF grant number 9983660 to ComeU
University.
HANDBOOK OF KNOT THEORY
Edited by William Menasco and Morwen Thistlethwaite
© 2005 Elsevier B.V. AU rights reserved
19
20 J.S. Birman and T.E. Brendle
4.6. Représentations of other mapping class groups 74
4.7. Additional représentations of ft, 75
5. The word and conjugacy problems in the braid groups 77
5.1. The Garside approach, as improved over the years 78
5.2. Generalizations: from B , to Garside groups 84
5.3. The new présentation and multiple Garside structures 86
5.4. Artin monoids and their groups 87
5.5. Braid groups and public key cryptography 88
5.6. The Nielsen Thurston approach to the conjugacy problem in By, 89
5.7. Other solutions to the word problem 92
6. A potpourri of miscellaneous results 95
6.1. Centralizers of braids and roots of braids 96
6.2. Singular braids, the singular braid monoid, and the desingularization map 96
6.3. The Tits conjecture 97
6.4. Braid groups are torsion free: a new proof 97
Acknowledgements 9g
Appendix. Computer programs 9g
Références 0.9
CHAPTER 3
Legendrian and Transversal Knots
John B. Etnyre
University of Pennsyhania, Department of Mathematics, 209 South 33rd Street,
Philadelphie PA 19104 6395, USA
E mail: etnyre@math.upenn.edu
Contents
1. Introduction 107
2. Définitions and examples
2.1. The standard contact structure on R1
1 OR
2.2. Other contact structures
2.3. Legendrian knots 10
2.4. Transverse knots
119
2.5. Types of classification
2.6. Invariants of Legendrian and transversal knots
2.7. Stabilizations 124
2.8. Surfaces and the classical invariants
2.9. Relation between Legendrian and transversal knots
3. Tightness and bounds on invariants
3.1. Bennequin s inequality
3.2. Slice genus , .
3.3. Other inequalities in (R i,,j) ,
4. New invariants .,,
4.1. Contact homology (aka Chekanov Eliashberg DGA)
4.2. Linearization ...
4.3. The characteristic algebra ]45
4.4. Lifting the DGA to I t,t ] ]4?
4.5. DGA s in the front projection 15Q
4.6. Décomposition invariants . ,
5. Classification results ,ca
5.1. The unknot 155
5.2. Torus knots ]57
5.3. Figure eight knot j^g
5.4. Connected sums i^q
5.5. Cables jg2
5.6. Links 164
5.7. The homotopy type of the space of Legendrian knots
106 J.B. Etnyre
5.8. Transverse knots 165
5.9. Knots in overtwisted contact structures 166
6. Higher dimensions 168
6.1. Legendrian knots in R2 +1 168
6.2. Generalizations of the Chekanov Eliashberg DGA 171
6.3. Examples 172
7. Applications 175
7.1. Legendrian surgery 175
7.2. Invariants of contact structures 176
7.3. Plane curves 177
7.4. Knot concordance 178
7.5. Invariants of classical knots 180
7.6. Contact homology and topological knot invariants 180
Références 182
CHAPTER 4
Knot Spinning
Greg Friedman
Department of Mathematics, Yale University, 10 Hillhouse Ave/P.O. Box 208283,
NewHaven, CT 06520 8283, USA
E mail: friedman@math.yale.edu
Contents
1. Introduction l89
2. Some basics 190
2.1. What is a knot? 19°
2.2. Knot équivalence l91
2.3. The unknot and toroidal décompositions of S
2.4. A useful excision
3. Basic spinnings
3.1. Simple spinning
3.2. Superspinning J9^
3.3. Frame spinning
4. Spinning with a twist
4.1. Twist spinning
4.2. Frame twist spinning 1
5. More gênerai spinnings
5.1. Deform spinning 1
5.2. Frame deform spinning ,
6. Other constructions
Références
CHAPTER 5
The Enumeration and Classification of Knots
and Links
Jim Hoste
Pitzer Collège, Department of Mathematics, 1050 N Mills Avenue,
Claremont. CA 91711. USA
E mail: jhoste@pitzer.edu
Contents
1. Introduction 211
2. Définitions 211
3. Classifying knots and links 214
4. Producing link tables 218
4.1. Encoding link diagrams 219
4.2. Generating ail alternating diagrams 223
4.3. Generating the nonalternating diagrams 227
5. Conclusion 228
Acknowledgements 229
Références 230
Abstract
The theoretical and practical aspects of link classification are described. with spécial
emphasis on the mathematics involved in récent, large scale link tabulations.
CHAPTER 6
Knot Diagrammatics
Louis H. Kauffman
Department of Mathematics, Statistics and Computer Science, University of Illinois,
851 South Morgan Street, Chicago, IL 60607 7045, USA
E mail: kauffinan@uic.edu
Contents
1. Introduction 235
2. Reidemeister moves 235
2.1. Reidemeister s theorem 238
2.2. Graph embeddings 241
3. Vassiliev invariants and invariants of rigid vertex graphs 246
3.1. Lie algebra weights 251
3.2. Vassiliev invariants and Witten s functional intégral 255
3.3. Combinatorial constructions for Vassiliev invariants 261
3.4. 8I7 264
4. Quantum link invariants 265
4.1. Knot amplitudes 265
4.2. Oriented amplitudes 269
4.3. Quantum link invariants and Vassiliev invariants 271
4.4. Vassiliev invariants and infinitésimal braiding 272
4.5. Weight Systems and the classical Yang Baxter équation 274
5. Hopf algebras and invariants of three manifolds 275
6. Temperley Lieb algebra 281
6.1. Parenthèses 284
7. Virtual knot theory 287
7.1. Fiat virtual knots and links 289
7.2. Interprétation of virtuals as stable classes of links in thickened surfaces 290
7.3. Jones polynomial of virtual knots 292
7.4. Biquandles 295
7.5. The Alexander biquandle 298
7.6. A Quantum model for GK(s, t), oriented and bi oriented quantum algebras 299
7.7. Invariants of three manifolds 301
7.8. Gauss diagrams and Vassiliev invariants 302
8. Other invariants 304
234 L.H. Kauffman
9. The bracket polynomial and the Jones polynomial 304
9.1. Thistlethwaite s example 307
9.2. Présent status of links not détectable by the Jones polynomial 308
9.3. Switching a crossing 311
Acknowledgements 313
Références 314
CHAPTER 7
A Survey of Classical Knot Concordance
Charles Livingston
Department of Mathematics, Indiana University, Bloominglon, IN 47405, USA
E mail: livingsl@indiana.edu
Contents
1. Introduction 321
2. Définitions 322
2.1. Knot theory and concordance 323
2.2. Algebraic concordance 323
3. Algebraic concordance invariants 325
3.1. Intégral invariants, signatures 326
3.2. The Arf invariant: Z2 326
3.3. Polynomial invariants: Z2 327
3.4. W(Q): Z2 and Z4 invariants 327
3.5. Quadratic polynomials 328
3.6. Other approaches to algebraic invariants 328
4. Casson Gordan invariants 329
4.1. Définitions 329
4.2. Main theorem 330
4.3. Invariants of W(C(r))®Q 330
5. Companionship and Casson Gordon invariants 331
5.1. Construction of companions 331
5.2. Casson Gordon invariants and companions 332
5.3. Genus one knots and the Seifert form 332
6. The topological category 334
6.1. Extensions 335
7. Smooth knot concordance 335
7.1. Further advances 336
8. Higher order obstructions and the filtration of C 336
9. Three dimensional knot properties and concordance 339
9.1. Primeness 339
9.2. Knot symmetry: amphicheirality 339
9.3. Reversibility and mutation 340
9.4. Periodicity 341
9.5. Genus 341
320 C. Livingston
9.6. Fibering 341
9.7. Unknotting number 342
10. Problems 342
Acknowledgements 344
Références 344
CHAPTER8
Knot Theory of Complex Plane Curves
Lee Rudolph*
Department of Mathematics and Computer Science and Department of Psychology,
Clark University, Worcester MA 01610 USA
E mail: lrudolph@black.clarku.edu
Contents ^51
1. Foreword ^^1
2. Preliminaries 352
2.1. Sets and groups j53
2.2. Spaces 356
2.3. Smooth maps 362
2.4. Knots, links, and Seifert surfaces %7
2.5. Framed links; Seifert forms 367
2.6. Fibered links, fiber surfaces, and open books ^
2.7. Polynomial invariants of knots and links .
2.8. Polynomial and analytic maps; algebraic and analytic sets 370
2.9. Configuration spaces and spaces of monic polynonuals J
2.10. Contact 3 manifolds, Stein domains, and Stem surfaces ^°
3. Braids and braided surfaces ^
3.1. Braid groups 37g
3.2. Géométrie braids and closed braids ^
3.3. Bands and espaliers •
34 Embeddedbandwords and braided Seifert surfaces •«
3.5. Plumbing and braided Seifert surfaces. •¦ •..••;, ,gfi
3.6. Labyrinths, braided surfaces in bidisks, and bra.ded nbbons w
4. Transverse C links .¦ .¦. , . . ion
4 1 Transverse C links are the same as quasipositive hnks £
4.2. Slice genus and unknotting number of transverse C links ^
4 3. Strongly quasipositive links 3gg
4.4. Non strongly quasipositive links
350 L. Rudolph
5. Complex plane curves in the small and in the large 399
5.1. Links of singularises as transverse C links 399
5.2. Links at infinity as transverse C links 400
6. Totally tangential C links 400
7. Relations to other research areas 402
7.1. Low dimensional real algebraic geometry; Hilbert s 16th problem 402
7.2. The Zariski Conjecture; knotgroups of complex plane curves 403
7.3. Keller s Jacobian Problem; embeddings and injections ofC in C~ 403
7.4. Chisini s statement; braid monodromy 404
7.5. Stein surfaces 404
8. The future of the knot theory of complex plane curves 405
8.1. Transverse C links and their Milnor maps 405
8.2. Transverse C links as links at infinity in the complex hyperbolic plane 405
8.3. Spaces of C links 406
8.4. Other questions 406
Acknowledgements 407
Appendix A. And now a few words from our inspirations 407
Références 409
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| id | DE-604.BV020872408 |
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| indexdate | 2025-02-03T17:28:01Z |
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| isbn | 044451452X |
| language | English |
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| spellingShingle | Handbook of knot theory Knot theory Knotentheorie (DE-588)4164318-5 gnd |
| subject_GND | (DE-588)4164318-5 |
| title | Handbook of knot theory |
| title_auth | Handbook of knot theory |
| title_exact_search | Handbook of knot theory |
| title_full | Handbook of knot theory ed. William Menasco ... |
| title_fullStr | Handbook of knot theory ed. William Menasco ... |
| title_full_unstemmed | Handbook of knot theory ed. William Menasco ... |
| title_short | Handbook of knot theory |
| title_sort | handbook of knot theory |
| topic | Knot theory Knotentheorie (DE-588)4164318-5 gnd |
| topic_facet | Knot theory Knotentheorie |
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