Mathematical Surveys and Monographs
Volume: 208;
2015;
410 pp;
Hardcover
MSC: Primary 57; 53;
Print ISBN: 978-1-4704-1737-6
Product Code: SURV/208
List Price: $110.00
AMS Member Price: $88.00
MAA Member Price: $99.00
Electronic ISBN: 978-1-4704-2739-9
Product Code: SURV/208.E
List Price: $65.00
AMS Member Price: $52.00
MAA Member Price: $58.50
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Supplemental Materials
Grid Homology for Knots and Links
Share this pagePeter S. Ozsváth; András I. Stipsicz; Zoltán Szabó
Knot theory is a classical area of low-dimensional
topology, directly connected with the theory of three-manifolds and
smooth four-manifold topology. In recent years, the subject has
undergone transformative changes thanks to its connections with a
number of other mathematical disciplines, including gauge theory;
representation theory and categorification; contact geometry; and the
theory of pseudo-holomorphic curves.
Starting from the
combinatorial point of view on knots using their grid diagrams, this
book serves as an introduction to knot theory, specifically as it
relates to some of the above developments. After a brief overview of
the background material in the subject, the book gives a
self-contained treatment of knot Floer homology from the point of view
of grid diagrams. Applications include computations of the unknotting
number and slice genus of torus knots (asked first in the 1960s and
settled in the 1990s), and tools to study variants of knot theory in
the presence of a contact structure. Additional topics are presented
to prepare readers for further study in holomorphic methods in
low-dimensional topology, especially Heegaard Floer homology.
The book could serve as a textbook for an advanced undergraduate or
part of a graduate course in knot theory. Standard background material
is sketched in the text and the appendices.
Readership
Graduate students and researchers interested in low-dimensional topology and geometry.
Table of Contents
Table of Contents
Grid Homology for Knots and Links
- Cover Cover11
- Title page iii4
- Contents vii8
- Chapter 1. Introduction 112
- Chapter 2. Knots and links in 𝑆³ 1324
- Chapter 3. Grid diagrams 4354
- Chapter 4. Grid homology 6576
- Chapter 5. The invariance of grid homology 91102
- Chapter 6. The unknotting number and 𝜏 113124
- Chapter 7. Basic properties of grid homology 127138
- Chapter 8. The slice genus and 𝜏 135146
- Chapter 9. The oriented skein exact sequence 151162
- Chapter 10. Grid homologies of alternating knots 167178
- Chapter 11. Grid homology for links 187198
- Chapter 12. Invariants of Legendrian and transverse knots 215226
- Chapter 13. The filtered grid complex 247258
- Chapter 14. More on the filtered chain complex 273284
- Chapter 15. Grid homology over the integers 291302
- Chapter 16. The holomorphic theory 325336
- Chapter 17. Open problems 339350
- Appendix A. Homological algebra 347358
- Appendix B. Basic theorems in knot theory 367378
- Bibliography 399410
- Index 407418
- Other titles in this series 411422
- Back Cover Back Cover1423