CiNii Books - 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

Table of 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. Knot Mosaics. 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 a Knot. 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 a Knotoid? 37. What Is a Braidoid? 38. What Is a Singular Knot? 39. Pseudoknots and Singular Knots. 40. An Introduction to the World 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 S and Other 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-Manifold. 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 Homolog. 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.

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Encyclopedia of knot theory

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