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El. knyga: Invitation to Knot Theory: Virtual and Classical [Taylor & Francis e-book]

(McKendree University, Lebanon, Illinois, USA)
  • Formatas: 288 pages, 23 Tables, black and white; 254 Illustrations, black and white
  • Išleidimo metai: 08-Mar-2016
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9781315370750
Kitos knygos pagal šią temą:
  • Taylor & Francis e-book
  • Kaina: 133,87 €*
  • * this price gives unlimited concurrent access for unlimited time
  • Standartinė kaina: 191,24 €
  • Sutaupote 30%
Invitation to Knot Theory: Virtual and ClassicalDidesnis paveikslėlis
  • Formatas: 288 pages, 23 Tables, black and white; 254 Illustrations, black and white
  • Išleidimo metai: 08-Mar-2016
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9781315370750
Kitos knygos pagal šią temą:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Partnerių interneto svetainė: https://www.taylorfrancis.com/books/9781315370750
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.
Section I Knots and crossings
Chapter 1 Virtual knots and links
3(16)
1.1 Curves In The Plane
3(4)
1.2 Virtual Links
7(8)
1.3 Oriented Virtual Link Diagrams
15(2)
1.4 Open Problems And Projects
17(2)
Chapter 2 Linking invariants
19(16)
2.1 Conditional Statements
19(4)
2.2 Writhe and Linking Number
23(4)
2.3 Difference Number
27(1)
2.4 Crossing Weight Numbers
28(5)
2.5 Open Problems and Projects
33(2)
Chapter 3 A multiverse of knots
35(14)
3.1 Flat And Free Links
35(8)
3.2 Welded, Singular, And Pseudo Knots
43(3)
3.3 New Knot Theories
46(1)
3.4 Open Problems And Projects
47(2)
Chapter 4 Crossing invariants
49(14)
4.1 Crossing Numbers
49(6)
4.2 Unknotting Numbers
55(4)
4.3 Unknotting Sequence Numbers
59(2)
4.4 Open Problems And Projects
61(2)
Chapter 5 Constructing knots
63(16)
5.1 Symmetry
63(4)
5.2 Tangles, Mutation, And Periodic Links
67(5)
5.3 Periodic Links And Satellite Knots
72(2)
5.4 Open Problems And Projects
74(5)
Section II Knot polynomials
Chapter 6 The bracket polynomial
79(16)
6.1 The Normalized Kauffman Bracket Polynomial
79(5)
6.2 The State Sum
84(2)
6.3 The Image Of The f-Polynomial
86(6)
6.4 Open Problems And Projects
92(3)
Chapter 7 Surfaces
95(20)
7.1 Surfaces
95(9)
7.2 Constructions Of Virtual Links
104(5)
7.3 Genus Of A Virtual Link
109(3)
7.4 Open Problems And Projects
112(3)
Chapter 8 Bracket polynomial II
115(12)
8.1 States And The Boundary Property
115(6)
8.2 Proper States
121(1)
8.3 Diagrams With One Virtual Crossing
122(2)
8.4 Open Problems And Projects
124(3)
Chapter 9 The checkerboard framing
127(12)
9.1 Checkerboard Framings
127(6)
9.2 Cut Points
133(2)
9.3 Extending The Theorem
135(3)
9.4 Open Problems And Projects
138(1)
Chapter 10 Modifications of the bracket polynomial
139(18)
10.1 The Flat Bracket
139(2)
10.2 The Arrow Polynomial
141(9)
10.3 Vassiliev Invariants
150(3)
10.4 Open Problems And Projects
153(4)
Section III Algebraic structures
Chapter 11 Quandles
157(12)
11.1 Tricoloring
157(3)
11.2 Quandles
160(3)
11.3 Knot Quandles
163(4)
11.4 Open Problems And Projects
167(2)
Chapter 12 Knots and quandles
169(16)
12.1 A Little Linear Algebra And The Trefoil
169(3)
12.2 The Determinant Of A Knot
172(8)
12.3 The Alexander Polynomial
180(2)
12.4 The Fundamental Group
182(1)
12.5 Open Problems And Projects
183(2)
Chapter 13 Biquandles
185(14)
13.1 The Biquandle Structure
185(5)
13.2 The Generalized Alexander Polynomial
190(6)
13.3 Open Problems And Projects
196(3)
Chapter 14 Gauss diagrams
199(14)
14.1 Gauss Words And Diagrams
199(6)
14.2 Parity And Parity Invariants
205(5)
14.3 Crossing Weight Number
210(2)
14.4 Open Problems And Projects
212(1)
Chapter 15 Applications
213(8)
15.1 Quantum Computation
213(2)
15.2 Textiles
215(3)
15.3 Open Problems And Projects
218(3)
Appendix A Tables
221(18)
A.1 Knot Tables
222(9)
A.2 Knot Invariants
231(8)
Appendix B References by chapter
239(14)
B.1
Chapter 1
239(1)
B.2
Chapter 2
240(1)
B.3
Chapter 3
240(1)
B.4
Chapter 4
241(2)
B.5
Chapter 5
243(1)
B.6
Chapter 6
244(1)
B.7
Chapter 7
245(1)
B.8
Chapter 8
245(1)
B.9
Chapter 9
246(1)
B.10
Chapter 10
246(2)
B.11
Chapter 11
248(1)
B.12
Chapter 12
249(1)
B.13
Chapter 13
250(1)
B.14
Chapter 14
250(1)
B.15
Chapter 15
251(2)
Index 253
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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