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An Invitation to Knot Theory
Virtual and Classical




ISBN 9781498701648
Published March 8, 2016 by Chapman and Hall/CRC
256 Pages 254 B/W Illustrations

 
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Book Description

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory

An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra.

The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

Table of Contents

Knots and crossings
Virtual knots and links
CURVES IN THE PLANE
VIRTUAL LINKS
ORIENTED VIRTUAL LINK DIAGRAMS

Linking invariants
CONDITIONAL STATEMENTS
WRITHE AND LINKING NUMBER
DIFFERENCE NUMBER
CROSSING WEIGHT NUMBERS

A multiverse of knots
FLAT AND FREE LINKS
WELDED, SINGULAR, AND PSEUDO KNOTS
NEW KNOT THEORIES

Crossing invariants
CROSSING NUMBERS
UNKNOTTING NUMBERS
UNKNOTTING SEQUENCE NUMBERS

Constructing knots
SYMMETRY
TANGLES, MUTATION, AND PERIODIC LINKS
PERIODIC LINKS AND SATELLITE KNOTS

Knot polynomials
The bracket polynomial

THE NORMALIZED KAUFFMAN BRACKET POLYNOMIAL
THE STATE SUM
THE IMAGE OF THE F-POLYNOMIAL

Surfaces
SURFACES
CONSTRUCTIONS OF VIRTUAL LINKS
GENUS OF A VIRTUAL LINK

Bracket polynomial II
STATES AND THE BOUNDARY PROPERTY
PROPER STATES
DIAGRAMS WITH ONE VIRTUAL CROSSING

The checkerboard framing
CHECKERBOARD FRAMINGS
CUT POINTS
EXTENDING THE KAUFFMAN-MURASUGI-THISTLETHWAITE THEOREM

Modifications of the bracket polynomial
THE FLAT BRACKET
THE ARROW POLYNOMIAL
VASSILIEV INVARIANTS

Algebraic structures
Quandles
TRICOLORING
QUANDLES
KNOT QUANDLES

Knots and quandles
A LITTLE LINEAR ALGEBRA AND THE TREFOIL
THE DETERMINANT OF A KNOT
THE ALEXANDER POLYNOMIAL
THE FUNDAMENTAL GROUP

Biquandles
THE BIQUANDLE STRUCTURE
THE GENERALIZED ALEXANDER POLYNOMIAL

Gauss diagrams
GAUSS WORDS AND DIAGRAMS
PARITY AND PARITY INVARIANTS
CROSSING WEIGHT NUMBER

Applications
QUANTUM COMPUTATION
TEXTILES

Appendix A: Tables
Appendix B: References by Chapter

Open problems and projects appear at the end of each chapter.

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Author(s)

Biography

Heather A. Dye is an associate professor of mathematics at McKendree University in Lebanon, Illinois, where she teaches linear algebra, probability, graph theory, and knot theory. She has published articles on virtual knot theory in the Journal of Knot Theory and its Ramifications, Algebraic and Geometric Topology, and Topology and its Applications. She is a member of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA).

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Author - Heather Ann Dye
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Heather Ann Dye

Associate Professor of Mathematics, McKendree University
Lebanon, IL, 62254

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Reviews

"This text provides an excellent entry point into virtual knot theory for undergraduates. Beginning with few prerequisites, the reader will advance to master the combinatorial and algebraic techniques that are most often employed in the literature. A student-centered book on the multiverse of knots (i.e., virtual knots, flat knots, free knots, welded knots, and pseudo knots) has long been awaited. The text aims not only to advertise recent developments in the field but to bring students to a point where they can begin thinking about interesting problems on their own. Each chapter contains not only exercises but projects, lists of open problems, and a carefully curated reading list. Students preparing to embark on an undergraduate research project in knot theory or virtual knot theory will greatly benefit from reading this well-written book!"
—Micah Chrisman, Ph.D., Associate Professor, Monmouth University

"This book will be greatly helpful and perfect for undergraduate and graduate students to study knot theory and see how ideas and techniques of mathematics learned at colleges or universities are used in research. Virtual knots are a hot topic in knot theory. By comparing virtual with classical, the book enables readers to understand the essence more easily and clearly."
—Seiichi Kamada, Vice-Director of Osaka City University Advanced Mathematical Institute and Professor of Mathematics, Osaka City University

"This is an excellent and well-organized introduction to classical and virtual knot theory that makes these subjects accessible to interested persons who may be unacquainted with point set topology or algebraic topology. The prerequisites for reading the book are a familiarity with basic college algebra and then later some abstract algebra and a familiarity or willingness to work with graphs (in the sense of graph theory) and pictorial diagrams (for knots and links) that are related to graphs. With this much background the book develops related topological themes such as knot polynomials, surfaces and quandles in a self-contained and clear manner. The subject of virtual knot theory is relatively new, having been introduced by Kauffman and by Goussarov, Polyak and Viro around 1996. Virtual knot theory can be learned right along with classical knot theory, as this book demonstrates, and it is a current research topic as well. So this book, elementary as it is, brings the reader right up to the frontier of present work in the theory of knots. It is exciting that knot theory, like graph theory, affords this possibility of stepping forward into the creative unknown."
—Louis H. Kauffman, Professor of Mathematics, University of Illinois at Chicago