1st Edition
Encyclopedia of Knot Theory
"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject."
– Ed Witten, Recipient of the Fields Medal
"I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field."
– Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis
Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deep mathematical connections, and critical applications. Many of the articles in this book are accessible to undergraduates who are working on research or taking an advanced undergraduate course in knot theory. More advanced articles will be useful to graduate students working on a related thesis topic, to researchers in another area of topology who are interested in current results in knot theory, and to scientists who study the topology and geometry of biopolymers.
Features
- Provides material that is useful and accessible to undergraduates, postgraduates, and full-time researchers
- Topics discussed provide an excellent catalyst for students to explore meaningful research and gain confidence and commitment to pursuing advanced degrees
- Edited and contributed by top researchers in the field of knot theory
I Introduction and History of Knots
Chapter 1. Introduction to Knots
Lewis D. Ludwig
II Standard and Nonstandard Representations of Knots
Chapter 2. Link Diagrams
Jim Hoste
Chapter 3. Gauss Diagrams
Inga Johnson
Chapter 4. DT Codes
Heather M. Russell
Chapter 5. Knot Mosaics
Lewis D. Ludwig
Chapter 6. Arc Presentations of Knots and Links
Hwa Jeong Lee
Chapter 7. Diagrammatic Representations of Knots and Links as Closed Braids
Sofia Lambropoulou
Chapter 8. Knots in Flows
Michael C. Sullivan
Chapter 9. Multi-Crossing Number of Knots and Links
Colin Adams
Chapter 10. Complementary Regions of Knot and Link Diagrams
Colin Adams
Chapter 11. Knot Tabulation
Jim Hoste
III Tangles
Chapter 12. What Is a Tangle?
Emille Davie Lawrence
Chapter 13. Rational and Non-Rational Tangles
Emille Davie Lawrence
Chapter 14. Persistent Invariants of Tangles
Daniel S. Silver and Susan G. Williams
IV Types of Knots
Chapter 15. Torus Knots
Jason Callahan
Chapter 16. Rational Knots and Their Generalizations
Robin T. Wilson
Chapter 17. Arborescent Knots and Links
Francis Bonahon
Chapter 18. Satellite Knots
Jennifer Schultens
Chapter 19. Hyperbolic Knots and Links
Colin Adams
Chapter 20. Alternating Knots
William W. Menasco
Chapter 21. Periodic Knots
Swatee Naik
V Knots and Surfaces
Chapter 22. Seifert Surfaces and Genus
Mark Brittenham
Chapter 23. Non-Orientable Spanning Surfaces for Knots
Thomas Kindred
Chapter 24. State Surfaces of Links
Efstratia Kalfagianni
Chapter 25. Turaev Surfaces
Seungwon Kim and Ilya Kofman
VI Invariants Defined in Terms of Min and Max
Chapter 26. Crossing Numbers
Alexander Zupan
Chapter 27. The Bridge Number of a Knot
Jennifer Schultens
Chapter 28. Alternating Distances of Knots
Adam Lowrance
Chapter 29. Superinvariants of Knots and Links
Colin Adams
VII Other Knotlike Objects
Chapter 30. Virtual Knot Theory
Louis H. Kauffman
Chapter 31. Virtual Knots and Surfaces
Micah Chrisman
Chapter 32. Virtual Knots and Parity
Heather A. Dye and Aaron Kaestner
Chapter 33. Forbidden Moves, Welded Knots and Virtual Unknotting
Sam Nelson
Chapter 34. Virtual Strings and Free Knots
Nicolas Petit
Chapter 35. Abstract and Twisted Links
Naoko Kamada
Chapter 36. What Is a Knotoid?
Harrison Chapman
Chapter 37. What Is a Braidoid?
Neslihan Gugumcu
Chapter 38. What Is a Singular Knot?
Zsuzsanna Dancso
Chapter 39. Pseudoknots and Singular Knots
Inga Johnson
Chapter 40. An Introduction to the World of Legendrian and Transverse Knots
Lisa Traynor
Chapter 41. Classical Invariants of Legendrian and Transverse Knots
Patricia Cahn
Chapter 42. Ruling and Augmentation Invariants of Legendrian Knots
Joshua M. Sabloff
VIII Higher Dimensional Knot Theory
Chapter 43. Broken Surface Diagrams and Roseman Moves
J. Scott Carter and Masahico Saito
Chapter 44. Movies and Movie Moves
J. Scott Carter and Masahico Saito
Chapter 45. Surface Braids and Braid Charts
Seiichi Kamada
Chapter 46. Marked Graph Diagrams and Yoshikawa Moves
Sang Youl Lee
Chapter 47. Knot Groups
Alexander Zupan
Chapter 48. Concordance Groups
Kate Kearney
IX Spatial Graph Theory
Chapter 49. Spatial Graphs
Stefan Friedl and Gerrit Herrmann
Chapter 50. A Brief Survey on Intrinsically Knotted and Linked Graphs
Ramin Naimi
Chapter 51. Chirality in Graphs
Hugh Howards
Chapter 52. Symmetries of Graphs Embedded in Sᶟ and Other 3-Manifolds
Erica Flapan
Chapter 53. Invariants of Spatial Graphs
Blake Mellor
Chapter 54. Legendrian Spatial Graphs
Danielle O’Donnol
Chapter 55. Linear Embeddings of Spatial Graphs
Elena Pavelescu
Chapter 56. Abstractly Planar Spatial Graphs
Scott A. Taylor
X Quantum Link Invariants
Chapter 57. Quantum Link Invariants
D. N. Yetter
Chapter 58. Satellite and Quantum Invariants
H. R. Morton
Chapter 59. Quantum Link Invariants: From QYBE and Braided Tensor Categories
Ruth Lawrence
Chapter 60. Knot Theory and Statistical Mechanics
Louis H. Kauffman
XI Polynomial Invariants
Chapter 61. What Is the Kauffman Bracket?
Charles Frohman
Chapter 62. Span of the Kauffman Bracket and the Tait Conjectures
Neal Stoltzfus
Chapter 63. Skein Modules of 3-Manifold
Rhea Palak Bakshi, Jozef H. Przytycki and Helen Wong
Chapter 64. The Conway Polynomial
Sergei Chmutov
Chapter 65. Twisted Alexander Polynomials
Stefano Vidussi
Chapter 66. The HOMFLYPT Polynomial
Jim Hoste
Chapter 67. The Kauffman Polynomials
Jianyuan K. Zhong
Chapter 68. Kauffman Polynomial on Graphs
Carmen Caprau
Chapter 69. Kauffman Bracket Skein Modules of 3-Manifolds
Rhea Palak Bakshi, Jozef Przytycki and Helen Wong
XII Homological Invariants
Chapter 70. Khovanov Link Homology
Radmila Sazdanovic
Chapter 71. A Short Survey on Knot Floer Homology
Andras I. Stipsicz
Chapter 72. An Introduction to Grid Homology
Andras I. Stipsicz
Chapter 73. Categorification
Volodymyr Mazorchuk
Chapter 74. Khovanov Homology and the Jones Polynomial
Alexander N. Shumakovitch
Chapter 75. Virtual Khovanov Homology
William Rushworth
XIII Algebraic and Combinatorial Invariants
Chapter 76. Knot Colorings
Pedro Lopes
Chapter 77. Quandle Cocycle Invariants
J. Scott Carter
Chapter 78. Kei and Symmetric Quandles
Kanako Oshiro
Chapter 79. Racks, Biquandles and Biracks
Sam Nelson
Chapter 80. Quantum Invariants via Hopf Algebras and Solutions to the Yang-Baxter Equation
Leandro Vendramin
Chapter 81. The Temperley-Lieb Algebra and Planar Algebras
Stephen Bigelow
Chapter 82. Vassiliev/Finite Type Invariants
Sergei Chmutov and Alexander Stoimenow
Chapter 83. Linking Number and Milnor Invariants
Jean-Baptiste Meilhan
XIV Physical Knot Theory
Chapter 84. Stick Number for Knots and Links
Colin Adams
Chapter 85. Random Knots
Kenneth Millett
Chapter 86. Open Knots
Julien Dorier, Dimos Goundaroulis, Eric J. Rawdon, and Andrzej Stasiak
Chapter 87. Random and Polygonal Spatial Graphs
Kenji Kozai
Chapter 88. Folded Ribbon Knots in the Plane
Elizabeth Denne
XV Knots and Science
Chapter 89. DNA Knots and Links
Isabel K. Darcy
Chapter 90. Protein Knots, Links, and Non-Planar Graphs
Helen Wong
Chapter 91. Synthetic Molecular Knots and Links
Erica Flapan
Biography
Colin Adams, Louis H. Kauffman, Sam Nelson
"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject."
– Ed Witten, Recipient of the Fields Medal"I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field."
– Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis"An encyclopedia is expected to be comprehensive, and to include independent expository articles on many topics. The Encyclopedia of Knot Theory is all this. This book will be an excellent introduction to topics in the field of knot theory for advanced undergraduates, graduate students, and researchers interested in knots from many directions."
– MAA Reviews
"Knot theory is an area of mathematics that requires no introduction, and while this massive tome is certainly no introductory text, it does give a panoramic — and, well, encyclopaedic — view of this vast subject.[. . . ] A book with such an ambitious remit is bound to contain omissions and oddities. [. . .] But this is a small point compared to what has been achieved by this encyclopaedia, which would make a fine addition to any personal or departmental library, or to a departmental coffee table."
– London Mathematical SocietyThe Encyclopedia of Knot Theory is close to 1000 pages, and every section, article, paragraph, and sentence inspires the reader to want to learn more knot theory. A wonderful attribute of this text is the reference section at the end of each article as opposed to the end of the book. This allows readers to highlight different sources that will allow them to dive deeper into the topic of that section. [. . .] And while it is nearly impossible to include discussions of every branch of the knot theorytree, the editors made a great choice to focus on current topics showing how the area is still a living subject.
[. . .] As a knot theory enthusiast, I truly enjoyed reading about topics I was more familiar with while also exploring topics that were new to me. As an educator, I am excited to share this book with my students and encourage them to read more articles on the topics. Some of the articles in the book include thoughtful open questions for researchers in the field to enjoy, while also providing background for anyone new to knot theory research to use as a foundation. All in all, I loved this text.
– American Mathematical Monthly
