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Softcover ISBN:  9781470434427 
Product Code:  SURV/208.S 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470427399 
Product Code:  SURV/208.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470434427 
eBook ISBN:  9781470427399 
Product Code:  SURV/208.S.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 208; 2015; 410 ppMSC: Primary 57; 53
Knot theory is a classical area of lowdimensional topology, directly connected with the theory of threemanifolds and smooth fourmanifold 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 pseudoholomorphic 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 selfcontained 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 lowdimensional 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.
ReadershipGraduate students and researchers interested in lowdimensional topology and geometry.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Knots and links in $S^3$

Chapter 3. Grid diagrams

Chapter 4. Grid homology

Chapter 5. The invariance of grid homology

Chapter 6. The unknotting number and $\tau $

Chapter 7. Basic properties of grid homology

Chapter 8. The slice genus and $\tau $

Chapter 9. The oriented skein exact sequence

Chapter 10. Grid homologies of alternating knots

Chapter 11. Grid homology for links

Chapter 12. Invariants of Legendrian and transverse knots

Chapter 13. The filtered grid complex

Chapter 14. More on the filtered chain complex

Chapter 15. Grid homology over the integers

Chapter 16. The holomorphic theory

Chapter 17. Open problems


Additional Material

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Knot theory is a classical area of lowdimensional topology, directly connected with the theory of threemanifolds and smooth fourmanifold 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 pseudoholomorphic 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 selfcontained 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 lowdimensional 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.
Graduate students and researchers interested in lowdimensional topology and geometry.

Chapters

Chapter 1. Introduction

Chapter 2. Knots and links in $S^3$

Chapter 3. Grid diagrams

Chapter 4. Grid homology

Chapter 5. The invariance of grid homology

Chapter 6. The unknotting number and $\tau $

Chapter 7. Basic properties of grid homology

Chapter 8. The slice genus and $\tau $

Chapter 9. The oriented skein exact sequence

Chapter 10. Grid homologies of alternating knots

Chapter 11. Grid homology for links

Chapter 12. Invariants of Legendrian and transverse knots

Chapter 13. The filtered grid complex

Chapter 14. More on the filtered chain complex

Chapter 15. Grid homology over the integers

Chapter 16. The holomorphic theory

Chapter 17. Open problems