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.
Autorentext
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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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.