954 Pages 80 Color & 485 B/W Illustrations
    by Chapman & Hall

    954 Pages 80 Color & 485 B/W Illustrations
    by Chapman & Hall

    954 Pages 80 Color & 485 B/W Illustrations
    by Chapman & Hall

    "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.


    • 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



    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 Undergraduate 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 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 Society

    The 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