Grid homology for knots and links:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2015]
|
| Schriftenreihe: | Mathematical surveys and monographs
volume 208 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturangaben |
| Beschreibung: | x, 410 Seiten Illustrationen, Diagramme |
| ISBN: | 9781470417376 |
Internformat
MARC
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| 100 | 1 | |a Ozsváth, Peter Steven |d 1967- |e Verfasser |0 (DE-588)1081461268 |4 aut | |
| 245 | 1 | 0 | |a Grid homology for knots and links |c Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2015] | |
| 264 | 4 | |c © 2015 | |
| 300 | |a x, 410 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v volume 208 | |
| 500 | |a Literaturangaben | ||
| 650 | 4 | |a Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 / 2msc | |
| 650 | 7 | |a Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds |2 msc | |
| 650 | 7 | |a Manifolds and cell complexes ... Differential topology ... Floer homology |2 msc | |
| 650 | 7 | |a Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general |2 msc | |
| 650 | 4 | |a Knot theory | |
| 650 | 4 | |a Link theory | |
| 650 | 4 | |a Homology theory | |
| 650 | 4 | |a Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 | |
| 650 | 4 | |a Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds | |
| 650 | 4 | |a Manifolds and cell complexes ... Differential topology ... Floer homology | |
| 650 | 4 | |a Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general | |
| 700 | 1 | |a Stipsicz, András |d 1966- |e Verfasser |0 (DE-588)138530785 |4 aut | |
| 700 | 1 | |a Szabó, Zoltán |d 1965- |e Verfasser |0 (DE-588)1081462256 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-2739-9 |
| 830 | 0 | |a Mathematical surveys and monographs |v volume 208 |w (DE-604)BV000018014 |9 208 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028762033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 912 | |a ebook | ||
| 999 | |a oai:aleph.bib-bvb.de:BVB01-028762033 | ||
Datensatz im Suchindex
| _version_ | 1804175893357133824 |
|---|---|
| adam_text | Contents
Chapter 1. Introduction 1
1.1. Grid homology and the Alexander polynomial 1
1.2. Applications of grid homology 3
1.3. Knot Floer homology 5
1.4. Comparison with Khovanov homology 7
1.5. On notational conventions 7
1.6. Necessary background 9
1.7. The organization of this book 9
1.8. Acknowledgements 11
Chapter 2. Knots and links in S3 13
2.1. Knots and links 13
2.2. Seifert surfaces 20
2.3. Signature and the unknotting number 21
2.4. The Alexander polynomial 25
2.5. Further constructions of knots and links 30
2.6. The slice genus 32
2.7. The Goeritz matrix and the signature 37
Chapter 3. Grid diagrams 43
3.1. Planar grid diagrams 43
3.2. Toroidal grid diagrams 49
3.3. Grids and the Alexander polynomial 51
3.4. Grid diagrams and Seifert surfaces 56
3.5. Grid diagrams and the fundamental group 63
Chapter 4. Grid homology 65
4.1. Grid states 65
4.2. Rectangles connecting grid states 66
4.3. The bigrading on grid states 68
4.4. The simplest version of grid homology 72
4.5. Background on chain complexes 73
4.6. The grid chain complex GC~ 75
4.7. The Alexander grading as a winding number 82
4.8. Computations 86
4.9. Further remarks 90
Chapter 5. The invariance of grid homology 91
5.1. Commutation invariance 91
5.2. Stabilization invariance 99
vii
CONTENTS
viii
5.3. Completion of the invariance proof for grid homology 107
5.4. The destabilization maps, revisited 108
5.5. Other variants of the grid complex 110
5.6. On the holomorphic theory 110
5.7. Further remarks on stabilization maps 110
Chapter 6. The unknotting number and r 113
6.1. The definition of r and its unknotting estimate 113
6.2. Construction of the crossing change maps 115
6.3. The Milnor conjecture for torus knots 120
6.4. Canonical grid cycles and estimates on r 122
Chapter 7. Basic properties of grid homology 127
7.1. Symmetries of the simply blocked grid homology 127
7.2. Genus bounds 129
7.3. General properties of unblocked grid homology 130
7.4. Symmetries of the unblocked theory 132
Chapter 8. The slice genus and r 135
8.1. Slice genus bounds from r and their consequences 135
8.2. A version of grid homology for links 136
8.3. Grid homology and saddle moves 139
8.4. Adding unknots to a link 143
8.5. Assembling the pieces: r bounds the slice genus 146
8.6. The existence of an exotic structure on R4 147
8.7. Slice bounds vs. unknotting bounds 149
Chapter 9. The oriented skein exact sequence 151
9.1. The skein exact sequence 151
9.2. The skein relation on the chain level 153
9.3. Proofs of the skein exact sequences 160
9.4. First computations using the skein sequence 162
9.5. Knots with identical grid homologies 163
9.6. The skein exact sequence and the crossing change map 164
9.7. Further remarks 166
Chapter 10. Grid homologies of alternating knots 167
10.1. Properties of the determinant of a link 167
10.2. The unoriented skein exact sequence 176
10.3. Grid homology groups for alternating knots 183
10.4. Further remarks 185
Chapter 11. Grid homology for links 187
11.1. The definition of grid homology for links 188
11.2. The Alexander multi-grading on grid homology 192
11.3. First examples 194
11.4. Symmetries of grid homology for links 196
11.5. The multi-variable Alexander polynomial 199
11.6. The Euler characteristic of multi-graded grid homology 203
11.7. Seifert genus bounds from grid homology for links 204
CONTENTS
IX
11.8. Further examples 205
11.9. Link polytopes and the Thurston norm 210
Chapter 12. Invariants of Legendrian and transverse knots 215
12.1. Legendrian knots in R3 216
12.2. Grid diagrams for Legendrian knots 220
12.3. Legendrian grid invariants 223
12.4. Applications of the Legendrian invariants 228
12.5. Transverse knots in R3 231
12.6. Applications of the transverse invariant 236
12.7. Invariants of Legendrian and transverse links 240
12.8. Transverse knots, grid diagrams, and braids 244
12.9. Further remarks 245
Chapter 13. The filtered grid complex 247
13.1. Some algebraic background 247
13.2. Defining the invariant 252
13.3. Topological invariance of the filtered quasi-isomorphism type 254
13.4. Filtered homotopy equivalences 268
Chapter 14. More on the filtered chain complex 273
14.1. Information in the filtered grid complex 273
14.2. Examples of filtered grid complexes 278
14.3. Refining the Legendrian and transverse invariants: definitions 281
14.4. Applications of the refined Legendrian and transverse invariants 285
14.5. Filtrations in the case of links 287
14.6. Remarks on three-manifold invariants 289
Chapter 15. Grid homology over the integers 291
15.1. Signs assignments and grid homology over Z 291
15.2. Existence and uniqueness of sign assignments 295
15.3. The invariance of grid homology over Z 302
15.4. Invariance in the filtered theory 308
15.5. Other grid homology constructions over Z 319
15.6. On the r-invariant 321
15.7. Relations in the spin group 321
15.8. Further remarks 323
Chapter 16. The holomorphic theory 325
16.1. Heegaard diagrams 325
16.2. From Heegaard diagrams to holomorphic curves 327
16.3. Multiple basepoints 333
16.4. Equivalence of knot Floer homology with grid homology 335
16.5. Further remarks 338
Chapter 17. Open problems 339
17.1. Open problems in grid homology 339
17.2. Open problems in knot Floer homology 341
Appendix A. Homological algebra 347
A.l. Chain complexes and their homology 347
X
CONTENTS
A. 2. Exact sequences 350
A.3. Mapping cones 352
A.4. On the structure of homology 357
A.5. Dual complexes 359
A.6. On filtered complexes 362
A. 7. Small models for filtered grid complexes 363
A. 8. Filtered quasi-isomorphism versus filtered homotopy type 365
Appendix B. Basic theorems in knot theory 367
B. l. The Reidemeister Theorem 367
B.2. Reidemeister moves in contact knot theory 373
B.3. The Reidemeister-Singer Theorem 382
B.4. Cromwell’s Theorem 387
B.5. Normal forms of cobordisms between knots 394
Bibliography 399
Index 407
|
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| spelling | Ozsváth, Peter Steven 1967- Verfasser (DE-588)1081461268 aut Grid homology for knots and links Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó Providence, Rhode Island American Mathematical Society [2015] © 2015 x, 410 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs volume 208 Literaturangaben Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 / 2msc Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds msc Manifolds and cell complexes ... Differential topology ... Floer homology msc Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general msc Knot theory Link theory Homology theory Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds Manifolds and cell complexes ... Differential topology ... Floer homology Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general Stipsicz, András 1966- Verfasser (DE-588)138530785 aut Szabó, Zoltán 1965- Verfasser (DE-588)1081462256 aut Erscheint auch als Online-Ausgabe 978-1-4704-2739-9 Mathematical surveys and monographs volume 208 (DE-604)BV000018014 208 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028762033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ozsváth, Peter Steven 1967- Stipsicz, András 1966- Szabó, Zoltán 1965- Grid homology for knots and links Mathematical surveys and monographs Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 / 2msc Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds msc Manifolds and cell complexes ... Differential topology ... Floer homology msc Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general msc Knot theory Link theory Homology theory Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds Manifolds and cell complexes ... Differential topology ... Floer homology Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general |
| title | Grid homology for knots and links |
| title_auth | Grid homology for knots and links |
| title_exact_search | Grid homology for knots and links |
| title_full | Grid homology for knots and links Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó |
| title_fullStr | Grid homology for knots and links Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó |
| title_full_unstemmed | Grid homology for knots and links Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó |
| title_short | Grid homology for knots and links |
| title_sort | grid homology for knots and links |
| topic | Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 / 2msc Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds msc Manifolds and cell complexes ... Differential topology ... Floer homology msc Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general msc Knot theory Link theory Homology theory Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds Manifolds and cell complexes ... Differential topology ... Floer homology Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general |
| topic_facet | Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 / 2msc Manifolds and cell complexes ... Low-dimensional topology ... Invariants of knots and 3-manifolds Manifolds and cell complexes ... Differential topology ... Floer homology Differential geometry ... Symplectic geometry, contact geometry ... Contact manifolds, general Knot theory Link theory Homology theory Manifolds and cell complexes ... Low-dimensional topology ... Knots and links in $S 3 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028762033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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