Encyclopedia of Knot Theory

Encyclopedia of Knot Theory is a collection of introductory articles on a wide range of topics in knot theory. The articles are all expository, each giving a short taste of a deeper subject. The topics range from classical, beginning with a brief history of mathematical knot theory, to very modern, including introductions to knot invariants that are the subject of recent research papers. The articles cover algebraic, geometric, and quantum tools used in knot theory, and applications of knots to other fields, from algebra to microbiology. In addition to giving background, articles frequently include an introduction to some of the relevant open problems in the field, and motivation for taking knot theory in that direction. They also include several references to relevant literature, for further reading and study. 

 

I believe the articles in this book will be an excellent resource for a broad range of readers. Several of the articles are elementary, and will be easily accessible to undergraduate students, for example, before embarking on related undergraduate research. Other articles require more sophisticated mathematical background, suitable for graduate students; these are typically situated in the book after a few more introductory articles on relevant topics. I also expect the book to be an excellent resource for research mathematicians, as it gives background on wider areas of knot theory than what is typically encountered in one corner of research.

 

The book is divided into fifteen parts, with each part containing several related articles. These fifteen parts are quite comprehensive, at least touching on what I believe are many of the major themes in knot theory.

 

The first four parts include history, descriptions of knots (diagrams, DT codes, flows, and multi-crossings, among others), tangles, and types of knots, i.e. families of related knots. These families include some of the most important classes of knots that appear in the literature: torus knots, hyperbolic knots, alternating knots, and others.

 

Part five involves surfaces that appear in knot complements, such as classical Seifert surfaces, but also state surfaces and Turaev surfaces that are more modern. Part six concerns invariants defined in terms of min and max, including classical crossing numbers and bridge numbers. Part seven concerns objects that are not knots and links, but very closely related, and whose study is heavily influenced by classical knot theory: virtual knots, welded knots, knotoids, and Legendrian knots. The book then turns to knot theory in higher dimensions, and then to spatial graph theory.

 

Parts ten through thirteen involve invariants: quantum, polynomial, homological, and algebraic and combinatorial invariants. Such invariants have played important roles in mathematical research for many years. For example, the homological invariants section gives introductions to Khovanov homology and knot Floer homology, both of which helped to resolve several open problems in knot theory earlier this century, and are objects of current study.

 

The book concludes with a section on physical knots, including an article on random knots and different ways to generate them, and a section on knots in science, with introductory articles on how knot theory is beginning to play roles in the study of DNA, proteins, and synthetic molecules.

 

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. 

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Jessica Purcell is a professor of mathematics at Monash University in Australia.  Her research is in the area of hyperbolic geometry and knot theory.