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Encyclopedia of Knot Theory

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"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

Author(s): Colin Adams (editor), Erica Flapan (editor), Allison Henrich (editor), Louis H. Kauffman (editor), Lewis D. Ludwig (editor), Sam Nelson (editor)
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2020

Language: English
Pages: 941

Cover
Half Title
Title Page
Copyright Page
Contents
I: Introduction and History of Knots
1. Introduction to Knots
II: Standard and Nonstandard Representations of Knots
2. Link Diagrams
3. Gauss Diagrams
4. DT Codes
5. KnotMosaics
6. Arc Presentations of Knots and Links
7. Diagrammatic Representations of Knots and Links as Closed Braids
8. Knots in Flows
9. Multi-Crossing Number of Knots and Links
10. Complementary Regions of Knot and Link Diagrams
11. Knot Tabulation
III: Tangles
12. What Is a Tangle?
13. Rational and Non-Rational Tangles
14. Persistent Invariants of Tangles
IV: Types of Knots
15. Torus Knots
16. Rational Knots and Their Generalizations
17. Arborescent Knots and Links
18. Satellite Knots
19. Hyperbolic Knots and Links
20. Alternating Knots
21. Periodic Knots
V: Knots and Surfaces
22. Seifert Surfaces and Genus
23. Non-Orientable Spanning Surfaces for Knots
24. State Surfaces of Links
25. Turaev Surfaces
VI: Invariants Defined in Terms of Min and Max
26. Crossing Numbers
27. The Bridge Number of aKnot
28. Alternating Distances of Knots
29. Superinvariants of Knots and Links
VII: Other Knotlike Objects
30. Virtual Knot Theory
31. Virtual Knots and Surfaces
32. Virtual Knots and Parity
33. Forbidden Moves, Welded Knots and Virtual Unknotting
34. Virtual Strings and Free Knots
35. Abstract and Twisted Links
36. What Is aKnotoid?
37. What Is a Braidoid?
38. What Is a Singular Knot?
39. Pseudoknots and Singular Knots
40. An Introduction to theWorld of Legendrian and Transverse Knots
41. Classical Invariants of Legendrian and Transverse Knots
42. Ruling and Augmentation Invariants of Legendrian Knots
VIII: Higher Dimensional Knot Theory
43. Broken Surface Diagrams and Roseman Moves
44. Movies and Movie Moves
45. Surface Braids and Braid Charts
46. Marked Graph Diagrams and Yoshikawa Moves
47. Knot Groups
48. Concordance Groups
IX: Spatial Graph Theory
49. Spatial Graphs
50. A Brief Survey on Intrinsically Knotted and Linked Graphs
51. Chirality in Graphs
52. Symmetries of Graphs Embedded in S3 andOther 3-Manifolds
53. Invariants of Spatial Graphs
54. Legendrian Spatial Graphs
55. Linear Embeddings of Spatial Graphs
56. Abstractly Planar Spatial Graphs
X: Quantum Link Invariants
57. Quantum Link Invariants
58. Satellite and Quantum Invariants
59. Quantum Link Invariants: From QYBE and Braided Tensor Categories
60. Knot Theory and Statistical Mechanics
XI: Polynomial Invariants
61. What Is the Kauffman Bracket?
62. Span of the Kauffman Bracket and the Tait Conjectures
63. Skein Modules of 3-Manifolds
64. The Conway Polynomial
65. Twisted Alexander Polynomials
66. The HOMFLYPT Polynomial
67. The Kauffman Polynomials
68. Kauffman Polynomial on Graphs
69. Kauffman Bracket Skein Modules of 3-Manifolds
XII: Homological Invariants
70. Khovanov Link Homology
71. A Short Survey on Knot Floer Homology
72. An Introduction to Grid Homology
73. Categorification
74. Khovanov Homology and the Jones Polynomial
75. Virtual Khovanov Homology
XIII: Algebraic and Combinatorial Invariants
76. Knot Colorings
77. Quandle Cocycle Invariants
78. Kei and Symmetric Quandles
79. Racks, Biquandles and Biracks
80. Quantum Invariants via Hopf Algebras and Solutions to the Yang-Baxter Equation
81. The Temperley-Lieb Algebra and Planar Algebras
82. Vassiliev/Finite-Type Invariants
83. Linking Number and Milnor Invariants
XIV: Physical Knot Theory
84. Stick Number for Knots and Links
85. Random Knots
86. Open Knots
87. Random and Polygonal Spatial Graphs
88. Folded Ribbon Knots in the Plane
XV: Knots and Science
89. DNA Knots and Links
90. Protein Knots, Links and Non-Planar Graphs
91. Synthetic Molecular Knots and Links
Index