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# Encyclopedia of Knot Theory

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## Book Description

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 which is useful and accessible to undergraduates, post-graduates, 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 to by top researchers in the field of Knot Theory

## Table of Contents

**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 *3 and Other 3-Manifolds

Erica Flapan

Chapter 53. Invariants of Spatial Graphs

Blake Mellor

Chapter 54. 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

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

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

**XIII Algebraic and Combinatorial Invariants**

Chapter 74. Knot Colorings

Pedro Lopes

Chapter 75. Quandle Cocycle Invariants

J. Scott Carter

Chapter 76. Kei and Symmetric Quandles

Kanako Oshiro

Chapter 77. Racks, Biquandles and Biracks

Sam Nelson

Chapter 78. Quantum Invariants via Hopf Algebras and Solutions to the Yang-Baxter Equation

Leandro Vendramin

Chapter 79. The Temperley-Lieb Algebra and Planar Algebras

Stephen Bigelow

Chapter 80. Vassiliev/Finite Type Invariants

Sergei Chmutov and Alexander Stoimenow

Chapter 81. Linking Number and Milnor Invariants

Jean-Baptiste Meilhan

**XIV Physical Knot Theory**

Chapter 82. Stick Number for Knots and Links

Colin Adams

Chapter 83. Random Knots

Kenneth Millett

Chapter 84. Open Knots

Julien Dorier, Dimos Goundaroulis, Eric J. Rawdon, and Andrzej Stasiak

Chapter 85. Random and Polygonal Spatial Graphs

Kenji Kozai

Chapter 86. Folded Ribbon Knots in the Plane

Elizabeth Denne

**XV Knots and Science**

Chapter 87. DNA Knots and Links

Isabel K. Darcy

Chapter 88. Protein Knots, Links, and Non-Planar Graphs

Helen Wong

Chapter 89. Synthetic Molecular Knots and Links

*Erica Flapan*

## Editor(s)

### 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 publishedc*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 130cundergraduates oncoriginal 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.