1st Edition

# Encyclopedia of Knot Theory

**Also available as eBook on:**

"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. Kau*ff*man*

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. Sablo*ff

**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 *S*ᶟ 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. Kau*ff*man*

**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** is the Thomas T. Read Professor of Mathematics at Williams College, having received his Ph.D. from the University of Wisconsin-Madison in 1983 and his Bachelor of Science from MIT in 1978. He is the author or co-author of numerous research papers in knot theory and low-dimensional topology and nine books, including the text *The Knot Book*~~"~~, the comic book *Why Knot?* and the text *Introduction to Topology: Pure and Applied*. He is a managing editor for the *Journal of Knot Theory and its Ramifications* and an editor for the recently published *Knots, Low Dimensional Topology and Applications*, Springer, 2019. A recipient of the Haimo National Distinguished Teaching Award, he has also been an MAA Polya Lecturer, a Sigma Xi Distinguished Lecturer, and a recipient of the Robert Foster Cherry Teaching Award. He has worked with over 130 undergraduates on original research in knot theory and low-dimensional topology.

He is also the humor columnist for the expository math magazine, the *Mathematical Intelligencer.*

**Erica Flapan** was a professor at Pomona College from 1986 to 2018. From 2000 until 2014, Flapan taught at the Summer Mathematics Program for freshmen and sophomore women at Carleton College, and mentored many of those women as they got their PhD’s in mathematics. In 2011, Flapan won the Mathematical Association of America’s Haimo award for distinguished college or university teaching of mathematics. Then in 2012, she was selected as an inaugural fellow of the American Mathematical Society. From 2015-2017, she was a Polya Lecturer for the MAA. Since 2019, she has been the Editor in Chief of the Notices of the American Mathematical Society.

Erica Flapan has published extensively in topology and its applications to chemistry and molecular biology. In addition to her many research papers, she has published an article in the College Mathematics Journal entitled “How to be a good teacher is an undecidable problem,” as well as four books. Her first book, entitled `*When Topology Meets Chemistry* was published jointly by the Mathematical Association of America and Cambridge University Press. Her second book entitled `*Applications of Knot Theory*, is a collection of articles that Flapan co-edited with Professor Dorothy Buck of Imperial College London. Flapan also co-authored a textbook entitled `*Number Theory: A Lively Introduction with Proofs, Applications, and Stories* with James Pommersheim and Tim Marks, published by John Wiley and sons. Finally, in 2016, the AMS published her book entitled *Knots, Molecules, and the Universe: An Introduction to Topology*, which is aimed at first and second year college students.

**Allison Henrich** is a Professor of Mathematics at Seattle University. She earned her PhD in Mathematics from Dartmouth College in 2008 and her undergraduate degrees in Mathematics and Philosophy from the University of Washington in 2003. Allison has been dedicated to providing undergraduates with high-quality research experiences since she mentored her first group of students in the SMALL REU at Williams College in 2009, under the mentorship of Colin Adams. Since then, she has mentored 35 beginning researchers, both undergraduates and high school students. She has directed and mentored students in the SUMmER REU at Seattle University and has served as a councilor on the Council on Undergraduate Research. In addition, Allison co-authored *An Interactive Introduction to Knot Theory* for undergraduates interested in exploring knots with a researcher’s mindset, and she co-*authored A Mathematician’s Practical Guide to Mentoring Undergraduate Research*, a resource for math faculty. Allison is excited about the publication of the *Concise Encyclopedia of Knot Theory*, as it will be an essential resource for beginning knot theory researchers!

**Louis Kauffman** was born in Potsdam, New York on Feb. 3, 1945. As a teenager he developed interests in Boolean algebra, circuits, diagrammatic logic and experiments related to non-linear pendulum oscillations. He received the degree of B.S. in Mathematics from MIT in 1966 and went to Princeton University for graduate school where he was awarded PhD in 1972. At Princeton he studied with William Browder and began working with knots and with singularities of complex hypersurfaces through conversations with the knot theorist Ralph Fox and the lectures of F. Hirzebruch and J. Milnor. From January 1971 to May 2017, he taught at the University of Illinois at Chicago where he is now Emeritus Professor of Mathematics.

Kauffman's research is in algebraic topology and particularly in low dimensional topology (knot theory) and its relationships with algebra, logic, combinatorics, mathematical physics and natural science. He is particularly interested in the structure of formal diagrammatic systems such as the system of knot diagrams that is one of the foundational approaches to knot theory. Kauffman’s research in Virtual Knot Theory has opened a new field of knot theory and has resulted in the discovery of new invariants of knots and links. His work on Temperley-Lieb Recoupling Theory with Sostences Lins led to new approaches to the Fibonacci model (Kitaev) for topological quantum computing. He has recently worked on representations of the Artin braid group related to the structure of Majorana Fermions.

His interdisciplinary research on cybernetics is concentrated on foundational understanding of mathematics based in the act of distinction, recursion and eigenform (the general structure of fixed points). This research is important both for mathematics and for the connections that are revealed in cybernetics with many other fields of study including the understanding of cognition, language, social systems and natural science.

Kauffman is the author of four books on knot theory, a book on map coloring and the reformulation of mathematical problems, *Map Reformulation *(Princelet Editions; London and Zurich [1986]) and is the editor of the World-Scientific Book Series *On Knots and Everything*. He is the Editor in Chief and founding editor of the *Journal of Knot Theory and Its Ramifications*. He is the co-editor of the review volume (with R. Baadhio) *Quantum Topology* and editor of the review volume *Knots and Applications*, both published by World Scientific Press and other review volumes in the Series on Knots and Everything.

Kauffman is the recipient of a 1993 University Scholar Award by the University of Illinois at Chicago. He was also awarded Warren McCulloch Memorial Award of the American Society for Cybernetics for significant contributions to the field of Cybernetics in the same year. He has also been awarded the 1996 award of the Alternative Natural Philosophy Association for his contribution to the understanding of discrete physics and the 2014 Norbert Wiener Medal from the American Society for Cybernetics. He was the president of the American Society for Cybernetics from 2005-2008 and the former Polya Lecturer for MAA (2008-2010). He was elected a Fellow of the American Mathematical Society in 2014.

Kauffman writes a column “Virtual Logic” for the Journal *Cybernetics and Human Knowing* and he plays clarinet in the Chicago-based ChickenFat Klezmer Orchestra.

**Lewis D. Ludwig** is a professor of mathematics at Denison University. He holds a PhD and Master’s in Mathematics from Ohio University and Miami University. His research in the field of topology mainly focuses on knot theory, a study in which he engages undergraduate students; nearly a third of his publications involve undergraduate collaborators. Lew created and co-hosted the **Un**dergraduate **Knot** Theory **Conference, **the UnKnot Conference, at Denison University which cumulatively supported over 300 undergraduate students, their faculty mentors, and researchers. Lew was a Co-PI and Project Team coordinator for the MAA Instructional Practices Guide Project – a guide to evidence-based instructional practices in undergraduate mathematics. He was creator and senior-editor for Teaching Tidbits blog for the MAA, a blog which provided techniques on evidence- based active teaching strategies for mathematics. Lew won the Distinguished Teaching Award for the Ohio Section of the MAA in 2013 and held the Nancy Eshelman Brickman Endowed Professorship.

**Sam Nelson** is a Professor and Chair of the Department of Mathematical Sciences at Claremont McKenna College. He earned his PhD in Mathematics at Louisiana State University in 2002 and his BS in Mathematics with minor in Philosophy at the University of Wyoming in 1996. A member of UWyo's first cohort of McNair Scholars, he has published 75 papers including over 40 with undergraduate and high school student co-authors, and is co-author of *Quandles: An Introduction to the Algebra of Knots*, the first textbook on Quandle theory. He has served as the Section Chair of the SoCal-Nevada section of the Mathematical Association of America and is an active member of the Mathematical Society of Japan and the American Mathematical Society, where he has co-organized 17 special sessions on Algebraic Structures in Knot Theory. He sits on the editorial boards of the *Journal of Knot Theory and its Ramifications* and the *Communications of the Korean Mathematical Society*, is the recipient of two Simons Foundation Collaboration grants, and was recently honoured as the recipient of Claremont McKenna College's annual Faculty Research Award. When not doing mathematics, he creates electronic music from knot diagrams as independent artist and composer Modulo Torsion.

"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 Theoryis 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 Society

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.The Encyclopedia of Knot Theory[. . .] 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