An Invitation to Knot Theory: Virtual and Classical

An Invitation to Knot Theory: Virtual and Classical

by Heather A. Dye

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Product Details

ISBN-13: 9781498701648
Publisher: Taylor & Francis
Publication date: 03/08/2016
Pages: 286
Product dimensions: 7.00(w) x 10.00(h) x 0.80(d)

About the Author

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).

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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