An invitation to knot theory : virtual and classical / Heather A. Dye, McKendree University, Lebanon, Illinois, USA.

Format:

Book

Author/Creator:

Dye, Heather A., author.

Language:

English

Subjects (All):

Knot theory.

Physical Description:

xxx, 256 pages : illustrations ; 27 cm

Place of Publication:

Boca Raton : CRC Press, Taylor & Francis Group, [2016]

Summary:

An Invitation to Knot Theory: Virtual and Classical gives you a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for you to research knot theory and read journal articles on your 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. Features: Covers virtual knots and other versions of knot equivalence - the only knot theory textbook to do so, Describes how proof strategies are used, Incorporates many examples gathered directly from current researches as well as well-known applications and results for classical knots and links, Includes problem sets and suggestions for undergraduate projects in each chapter, Provides further reading recommendations based on peer-reviewed journals. Book jacket.

Contents:

Section I Knots and Crossings

Chapter 1 "Virtual Knots and Links 3

1.1 Curves in the Plane 3

1.2 Virtual Links 7

1.3 Oriented Virtual Link Diagrams 15

1.4 Open Problems and Projects 17

Chapter 2 Linking Invariants 19

2.1 Conditional Statements 19

2.2 Writhe And Linking Number 23

2.3 Difference Number 27

2.4 Crossing Weight Numbers 28

2.5 Open Problems and Projects 33

Chapter 3 A multiverse Of Knots 35

3.1 Flat and Free Links 35

3.2 Welded, Singular, And Pseudo Knots 43

3.3 New Knot Theories 46

3.4 Open Problems and Projects 47

Chapter 4 Crossing Invariants 49

4.1 Crossing Numbers 49

4.2 Unknotting Numbers 55

4.3 Unknotting Sequence Numbers 59

4.4 Open Problems and Projects 61

Chapter 5 Constructing Knots 63

5.1 Symmetry 63

5.2 Tangles, Mutation, and Periodic Links 67

5.3 Periodic Links and Satellite Knots 72

5.4 Open Problems and Projects 74

Section II Knot Polynomials

Chapter 6 The bracket polynomial 79

6.1 The Normalized Kauffman Bracket Polynomial 79

6.2 The State Sum 84

6.3 The Image of the F-Polynomial 86

6.4 Open Problems and Projects 92

Chapter 7 Surfaces 95

7.1 Surfaces 95

7.2 Constructions of Virtual Links 104

7.3 Genus of a Virtual Link 109

7.4 Open Problems and Projects 112

Chapter 8 Bracket Polynomial II 115

8.1 States and the Boundary Property 115

8.2 Proper States 121

8.3 Diagrams with One Virtual Crossing 122

8.4 Open Problems and Projects 124

Chapter 9 The checkerboard framing 127

9.1 Checkerboard Framings 127

9.2 Cut Points 133

9.3 Extending the Theorem 135

9.4 Open Problems and Projects 138

Chapter 10 Modifications of the bracket polynomial 139

10.1 The Flat Bracket 139

10.2 The Arrow Polynomial 141

10.3 Vassiliev Invariants 150

10.4 Open Problems and Projects 153

Section III Algebraic structures

Chapter 11 Quandles 157

11.1 Tricolorig 157

11.2 Quandles 160

11.3 Knot Quandles 163

11.4 Open Problems and Projects 167

Chapter 12 Knots and quandles 169

12.1 A Little Linear Algebra and The Trefoil 169

12.2 The Determinant of a Knot 172

12.3 The Alexander Polynomial 180

12.4 The Fundamental Group 182

12.5 Open Problems and Projects 183

Chapter 13 Biquandles 185

13.1 The Biquandle Structure 185

13.2 The Generalized Alexander Polynomial 190

13.3 Open Problems and Projects 196

Chapter 14 Gauss Diagrams 199

14.1 Gauss Words and Diagrams 199

14.2 Parity and Parity Invariants 205

14.3 Crossing Weight Number 210

14.4 Open Problems and Projects 212

Chapter 15 Applications 213

15.1 Quantum Computation 213

15.2 Textiles 215

15.3 Open Problems and Projects 218.

Notes:

"A Chapman & Hall Book."

Includes bibliographical references (pages 239-251) and index.

ISBN:

9781498701648

1498701647

OCLC:

920452534

Publisher Number:

99968479883