"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

part I|6 pages

Introduction and History of Knots

chapter 1|4 pages

Introduction to Knots

ByLewis D. Ludwig

part II|76 pages

Standard and Nonstandard Representations of Knots

chapter 2|4 pages

Link Diagrams

ByJim Hoste

chapter 3|4 pages

Gauss Diagrams

ByInga Johnson

chapter 4|4 pages

DT Codes

ByHeather M. Russell

chapter 5|6 pages

Knot Mosaics

ByLewis D. Ludwig

chapter 6|8 pages

Arc Presentations of Knots and Links

ByHwa Jeong Lee

chapter 8|10 pages

Knots in Flows

ByMichael C. Sullivan

chapter 9|8 pages

Multi-Crossing Number of Knots and Links

ByColin Adams

chapter 10|6 pages

Complementary Regions of Knot and Link Diagrams

ByColin Adams

chapter 11|6 pages

Knot Tabulation

ByJim Hoste

part III|20 pages


chapter 12|4 pages

What Is a Tangle?

ByEmille Davie Lawrence

chapter 13|8 pages

Rational and Non-Rational Tangles

ByEmille Davie Lawrence

chapter 14|6 pages

Persistent Invariants of Tangles

ByDaniel S. Silver, Susan G. Williams

part IV|82 pages

Types of Knots

chapter 15|6 pages

Torus Knots

ByJason Callahan

chapter 16|10 pages

Rational Knots and Their Generalizations

ByRobin T. Wilson

chapter 17|26 pages

Arborescent Knots and Links

ByFrancis Bonahon

chapter 18|12 pages

Satellite Knots

ByJennifer Schultens

chapter 19|8 pages

Hyperbolic Knots and Links

ByColin Adams

chapter 20|12 pages

Alternating Knots

ByWilliam W. Menasco

chapter 21|6 pages

Periodic Knots

BySwatee Naik

part Part V|36 pages

Knots and Surfaces

chapter 22|10 pages

Seifert Surfaces and Genus

ByMark Brittenham

chapter 23|8 pages

Non-Orientable Spanning Surfaces for Knots

ByThomas Kindred

chapter 24|8 pages

State Surfaces of Links

ByEfstratia Kalfagianni

chapter 25|8 pages

Turaev Surfaces

BySeungwon Kim, Ilya Kofman

part VI|38 pages

Invariants Defined in Terms of Min and Max

chapter 26|6 pages

Crossing Numbers

ByAlexander Zupan

chapter 27|14 pages

The Bridge Number of a Knot

ByJennifer Schultens

chapter 28|8 pages

Alternating Distances of Knots

ByAdam Lowrance

chapter 29|8 pages

Superinvariants of Knots and Links

ByColin Adams

part VII|152 pages

Other Knotlike Objects

chapter 30|50 pages

Virtual Knot Theory

ByLouis H. Kauffman

chapter 31|8 pages

Virtual Knots and Surfaces

ByMicah Chrisman

chapter 32|8 pages

Virtual Knots and Parity

ByHeather A. Dye, Aaron Kaestner

chapter 34|8 pages

Virtual Strings and Free Knots

ByNicolas Petit

chapter 35|8 pages

Abstract and Twisted Links

ByNaoko Kamada

chapter 36|8 pages

What Is a Knotoid?

ByHarrison Chapman

chapter 37|12 pages

What Is a Braidoid?

ByNeslihan Gügümcü

chapter 38|8 pages

What Is a Singular Knot?

ByZsuzsanna Dancso

chapter 39|6 pages

Pseudoknots and Singular Knots

ByInga Johnson

chapter 42|8 pages

Ruling and Augmentation Invariants of Legendrian Knots

ByJoshua M. Sabloff

part VIII|48 pages

Higher Dimensional Knot Theory

chapter 43|6 pages

Broken Surface Diagrams and Roseman Moves

ByJ. Scott Carter, Masahico Saito

chapter 44|6 pages

Movies and Movie Moves

ByJ. Scott Carter, Masahico Saito

chapter 45|8 pages

Surface Braids and Braid Charts

BySeiichi Kamada

chapter 46|10 pages

Marked Graph Diagrams and Yoshikawa Moves

BySang Youl Lee

chapter 47|8 pages

Knot Groups

ByAlexander Zupan

chapter 48|8 pages

Concordance Groups

ByKate Kearney

part IX|70 pages

Spatial Graph Theory

chapter 49|6 pages

Spatial Graphs

ByStefan Friedl, Gerrit Herrmann

chapter 51|8 pages

Chirality in Graphs

ByHugh Howards

chapter 53|12 pages

Invariants of Spatial Graphs

ByBlake Mellor

chapter 54|10 pages

Legendrian Spatial Graphs

ByDanielle O'Donnol

chapter 55|8 pages

Linear Embeddings of Spatial Graphs

ByElena Pavelescu

chapter 56|8 pages

Abstractly Planar Spatial Graphs

ByScott A. Taylor

part X|72 pages

Quantum Link Invariants

chapter 57|16 pages

Quantum Link Invariants

ByD. N. Yetter

chapter 58|12 pages

Satellite and Quantum Invariants

ByH. R. Morton

chapter 60|20 pages

Knot Theory and Statistical Mechanics

ByLouis H. Kauffman

part XI|66 pages

Polynomial Invariants

chapter 61|8 pages

What Is the Kauffman Bracket?

ByCharles Frohman

chapter 62|6 pages

Span of the Kauffman Bracket and the Tait Conjectures

ByNeal Stoltzfus

chapter 63|8 pages

Skein Modules of 3-Manifolds

ByRhea Palak Bakshi, Józef H. Przytycki, Helen Wong

chapter 64|6 pages

The Conway Polynomial

BySergei Chmutov

chapter 65|6 pages

Twisted Alexander Polynomials

ByStefano Vidussi

chapter 66|6 pages

The HOMFLYPT Polynomial

ByJim Hoste

chapter 67|6 pages

The Kauffman Polynomials

ByJianyuan K. Zhong

chapter 68|8 pages

Kauffman Polynomial on Graphs

ByCarmen Caprau

chapter 69|10 pages

Kauffman Bracket Skein Modules of 3-Manifolds

ByRhea Palak Bakshi, Józef H. Przytycki, Helen Wong

part XII|96 pages

Homological Invariants

chapter 70|12 pages

Khovanov Link Homology

ByRadmila Sazdanovic

chapter 71|20 pages

A Short Survey on Knot Floer Homology

ByAndrás I. Stipsicz

chapter 72|22 pages

An Introduction to Grid Homology

ByAndrás I. Stipsicz

chapter 73|10 pages


ByVolodymyr Mazorchuk

chapter 74|18 pages

Khovanov Homology and the Jones Polynomial

ByAlexander N. Shumakovitch

chapter 75|12 pages

Virtual Khovanov Homology

ByWilliam Rushworth

part XIII|68 pages

Algebraic and Combinatorial Invariants

chapter 76|8 pages

Knot Colorings

ByPedro Lopes

chapter 77|8 pages

Quandle Cocycle Invariants

ByJ. Scott Carter

chapter 78|8 pages

Kei and Symmetric Quandles

ByKanako Oshiro

chapter 79|6 pages

Racks, Biquandles and Biracks

BySam Nelson

chapter 81|6 pages

The Temperley-Lieb Algebra and Planar Algebras

ByStephen Bigelow

chapter 82|10 pages

Vassiliev/Finite-Type Invariants

BySergei Chmutov, Alexander Stoimenow

chapter 83|14 pages

Linking Number and Milnor Invariants

ByJean-Baptiste Meilhan

part XIV|68 pages

Physical Knot Theory

chapter 84|8 pages

Stick Number for Knots and Links

ByColin Adams

chapter 85|18 pages

Random Knots

ByKenneth Millett

chapter 86|18 pages

Open Knots

ByJulien Dorier, Dimos Goundaroulis, Eric J. Rawdon, Andrzej Stasiak

chapter 87|8 pages

Random and Polygonal Spatial Graphs

ByKenji Kozai

chapter 88|14 pages

Folded Ribbon Knots in the Plane

ByElizabeth Denne

part XV|28 pages

Knots and Science

chapter 89|10 pages

DNA Knots and Links

ByIsabel K. Darcy

chapter 90|8 pages

Protein Knots, Links and Non-Planar Graphs

ByHelen Wong

chapter 91|8 pages

Synthetic Molecular Knots and Links

ByErica Flapan