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# Encyclopedia of Knot Theory

DOI link for Encyclopedia of Knot Theory

Encyclopedia of Knot Theory book

# Encyclopedia of Knot Theory

DOI link for Encyclopedia of Knot Theory

Encyclopedia of Knot Theory book

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## ABSTRACT

"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

part Part I|6 pages

Introduction and History of Knots

entry 2Chapter 1|4 pages

Introduction to Knots

part Part II|76 pages

Standard and Nonstandard Representations of Knots

entry 8Chapter 2|4 pages

Link Diagrams

entry Chapter 3|4 pages

Gauss Diagrams

entry Chapter 4|4 pages

DT Codes

entry Chapter 5|6 pages

Knot Mosaics

entry Chapter 6|8 pages

Arc Presentations of Knots and Links

entry Chapter 7|18 pages

Diagrammatic Representations of Knots and Links as Closed Braids

entry Chapter 8|10 pages

Knots in Flows

entry Chapter 9|8 pages

Multi-Crossing Number of Knots and Links

entry Chapter 10|6 pages

Complementary Regions of Knot and Link Diagrams

entry Chapter 11|6 pages

Knot Tabulation

part Part III|20 pages

Tangles

entry 84Chapter 12|4 pages

What Is a Tangle?

entry Chapter 13|8 pages

Rational and Non-Rational Tangles

entry Chapter 14|6 pages

Persistent Invariants of Tangles

part Part IV|82 pages

Types of Knots

entry 104Chapter 15|6 pages

Torus Knots

entry Chapter 16|10 pages

Rational Knots and Their Generalizations

entry Chapter 17|26 pages

Arborescent Knots and Links

entry Chapter 18|12 pages

Satellite Knots

entry Chapter 19|8 pages

Hyperbolic Knots and Links

entry Chapter 20|12 pages

Alternating Knots

entry Chapter 21|6 pages

Periodic Knots

part Part V|36 pages

Knots and Surfaces

entry 186Chapter 22|10 pages

Seifert Surfaces and Genus

entry Chapter 23|8 pages

Non-Orientable Spanning Surfaces for Knots

entry Chapter 24|8 pages

State Surfaces of Links

entry Chapter 25|8 pages

Turaev Surfaces

part Part VI|38 pages

Invariants Defined in Terms of Min and Max

entry 222Chapter 26|6 pages

Crossing Numbers

entry Chapter 27|14 pages

The Bridge Number of a Knot

entry Chapter 28|8 pages

Alternating Distances of Knots

entry Chapter 29|8 pages

Superinvariants of Knots and Links

part Part VII|152 pages

Other Knotlike Objects

entry 260Chapter 30|50 pages

Virtual Knot Theory

entry Chapter 31|8 pages

Virtual Knots and Surfaces

entry Chapter 32|8 pages

Virtual Knots and Parity

entry Chapter 33|8 pages

Forbidden Moves, Welded Knots and Virtual Unknotting

entry Chapter 34|8 pages

Virtual Strings and Free Knots

entry Chapter 35|8 pages

Abstract and Twisted Links

entry Chapter 36|8 pages

What Is a Knotoid?

entry Chapter 37|12 pages

What Is a Braidoid?

entry Chapter 38|8 pages

What Is a Singular Knot?

entry Chapter 39|6 pages

Pseudoknots and Singular Knots

entry Chapter 40|8 pages

An Introduction to the World of Legendrian and Transverse Knots

entry Chapter 41|10 pages

Classical Invariants of Legendrian and Transverse Knots

entry Chapter 42|8 pages

Ruling and Augmentation Invariants of Legendrian Knots

part Part VIII|48 pages

Higher Dimensional Knot Theory

entry 412Chapter 43|6 pages

Broken Surface Diagrams and Roseman Moves

entry Chapter 44|6 pages

Movies and Movie Moves

entry Chapter 45|8 pages

Surface Braids and Braid Charts

entry Chapter 46|10 pages

Marked Graph Diagrams and Yoshikawa Moves

entry Chapter 47|8 pages

Knot Groups

entry Chapter 48|8 pages

Concordance Groups

part Part IX|70 pages

Spatial Graph Theory

entry 460Chapter 49|6 pages

Spatial Graphs

entry Chapter 50|10 pages

A Brief Survey on Intrinsically Knotted and Linked Graphs

entry Chapter 51|8 pages

Chirality in Graphs

entry Chapter 52|6 pages

Symmetries of Graphs Embedded in S 3 and Other 3-Manifolds

entry Chapter 53|12 pages

Invariants of Spatial Graphs

entry Chapter 54|10 pages

Legendrian Spatial Graphs

entry Chapter 55|8 pages

Linear Embeddings of Spatial Graphs

entry Chapter 56|8 pages

Abstractly Planar Spatial Graphs

part Part X|72 pages

Quantum Link Invariants

entry 530Chapter 57|16 pages

Quantum Link Invariants

entry Chapter 58|12 pages

Satellite and Quantum Invariants

entry Chapter 59|22 pages

Quantum Link Invariants: From QYBE and Braided Tensor Categories

entry Chapter 60|20 pages

Knot Theory and Statistical Mechanics

part Part XI|66 pages

Polynomial Invariants

entry 602Chapter 61|8 pages

What Is the Kauffman Bracket?

entry Chapter 62|6 pages

Span of the Kauffman Bracket and the Tait Conjectures

entry Chapter 63|8 pages

Skein Modules of 3-Manifolds

entry Chapter 64|6 pages

The Conway Polynomial

entry Chapter 65|6 pages

Twisted Alexander Polynomials

entry Chapter 66|6 pages

The HOMFLYPT Polynomial

entry Chapter 67|6 pages

The Kauffman Polynomials

entry Chapter 68|8 pages

Kauffman Polynomial on Graphs

entry Chapter 69|10 pages

Kauffman Bracket Skein Modules of 3-Manifolds

part Part XII|96 pages

Homological Invariants

entry 668Chapter 70|12 pages

Khovanov Link Homology

entry Chapter 71|20 pages

A Short Survey on Knot Floer Homology

entry Chapter 72|22 pages

An Introduction to Grid Homology

entry Chapter 73|10 pages

Categorification

entry Chapter 74|18 pages

Khovanov Homology and the Jones Polynomial

entry Chapter 74|12 pages

Virtual Khovanov Homology

part Part XIII|68 pages

Algebraic and Combinatorial Invariants

entry 764Chapter 76|8 pages

Knot Colorings

entry Chapter 77|8 pages

Quandle Cocycle Invariants

entry Chapter 78|8 pages

Kei and Symmetric Quandles

entry Chapter 79|6 pages

Racks, Biquandles and Biracks

entry Chapter 80|6 pages

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

entry Chapter 81|6 pages

The Temperley-Lieb Algebra and Planar Algebras

entry Chapter 82|10 pages

Vassiliev/Finite-Type Invariants

entry Chapter 83|14 pages

Linking Number and Milnor Invariants

part Part XIV|68 pages

Physical Knot Theory

entry 832Chapter 84|8 pages

Stick Number for Knots and Links

entry Chapter 85|18 pages

Random Knots

entry Chapter 86|18 pages

Open Knots

entry Chapter 87|8 pages

Random and Polygonal Spatial Graphs

entry Chapter 88|14 pages

Folded Ribbon Knots in the Plane

part Part XV|28 pages

Knots and Science

entry 900Chapter 89|10 pages

DNA Knots and Links

entry Chapter 90|8 pages

Protein Knots, Links and Non-Planar Graphs

entry Chapter 91|8 pages

Synthetic Molecular Knots and Links