Knots, molecules, and the universe: an introduction to topology
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2016]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xvii, 386 Seiten Illustrationen |
| ISBN: | 9781470425357 |
Internformat
MARC
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| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2016] | |
| 300 | |a xvii, 386 Seiten |b Illustrationen | ||
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| 650 | 4 | |a Topology |v Textbooks | |
| 650 | 4 | |a Algebraic topology |v Textbooks | |
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| 650 | 4 | |a Geometry |v Textbooks | |
| 650 | 4 | |a Molecular biology |v Textbooks | |
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Datensatz im Suchindex
| _version_ | 1822603543847108608 |
|---|---|
| adam_text |
Contents
Preface xi
Acknowledgments xv
Part 1- Universes
Chapter 1. An Introduction to the Shape of the Universe 3
1. To Infinity and Beyond 3
2. A. Square and His Universe 5
3. Straightest Paths in Flatland 8
4. Exploring the Shape of a Cave 11
5. Creating Universes by Gluing 13
6. Games on a Gluing Diagram for a Cylinder and a Torus 15
7. Extended Diagrams 17
8. Introducing the 3-Dimensional Torus and Sphere 19
9. Distinguishing a 2-Sphere from a 2-Torus 22
10. Distinguishing a 3-Sphere from a 3-Torus 24
11. Exercises 26
Chapter 2. Visualizing Four Dimensions 29
1. Teaching A. Square About the Third Dimension 29
2. Projections and Perceptions 31
3. Movies and Movement 33
4. Inductively Defining Cubes of Dimensions 0, 1, 2, and 3 36
5. A. Square Learns About a 3-Dimensional Cube 37
6. A 4-Dimensional Cube 40
7. Tetrahedra in Various Dimensions 42
8. Exercises 44
Chapter 3. Geometry and Topology of Different Universes 49
1. Intrinsic and Extrinsic Properties of a Space 49
2. Intrinsic and Extrinsic Geometry 52
3. Straightest Paths and Geodesics 53
4. The Definition of a Triangle 59
5. The Sum of the Angles of a Triangle 60
6. Triangles on a Flat and Curved Torus 61
7. Triangles on a Flat and a Curved Cylinder 62
8. Extrinsic and Intrinsic Topology 64
9. Using Loops to Understand Intrinsic Topology 67
10. Local and Global Properties 69
V
vi CONTENTS
11. Manifolds 71
12. Assumptions about Universes 72
13. A 2-Dimensional Sphere with a Point Removed 74
14. A 3-Dimensional Sphere with a Point Removed 77
15. Curves on a Torus 78
16. How to Draw a (p, q)-Curve on a Flat Torus 80
17. Lines in the Extended Diagram of a Flat Torus 83
18. Exercises 84
Chapter 4. Orientability 87
1. The Mobius Strip 87
2. Orientation Reversing Paths 89
3. The Klein Bottle 90
4. The 3-Dimensional Klein Bottle 92
5. Tic-Tac-Toe on a Torus and Klein Bottle 93
6. The Projective Plane 96
7. The Projective Plane with a Disk Removed 98
8. Projective 3-Dimensional Space 100
9. 1-Sided and 2-Sided Surfaces 101
10. Non-orient ability and l֊Sidedness 103
11. Exercises 106
Chapter 5. Flat Manifolds 109
1. Flat Surfaces 109
2. Flat Gluing Diagrams 110
3. The Point of a Cone 114
4. Using Extended Diagrams to Understand Cone Points 116
5. Anti-cone Points 119
6. We Show that the 3-Torus Is Flat 120
7. A Method to Determine if a Glued Up Cube Is Flat 122
8. Extended Diagrams of Glued Up Cubes 123
9. Other Types of Gluings of a Cube 127
10. Exercises 128
Chapter 6. Connected Sums of Spaces 135
1. Einstein-Rosen Bridges 135
2. Connected Sums of Surfaces 136
3. Arithmetic Properties of the Connected Sum 137
4. Gluing Diagrams for nT2 and nP2 138
5. The Classification of Surfaces 140
6. Dividing a Surface into Vertices, Edges, and Faces 142
7. The Euler Characteristic of a Surface 144
8. The Euler Characteristic of Connected Sums 146
9. The Genus of a Surface 147
10. The Genus of nT2 and nP2 150
11. Connected Sums of 3-Manifolds 151
12. Exercises 153
Chapter 7. Products of Spaces 157
1. Products of Sets 157
CONTENTS vii
2. Products of Spaces 159
3. A. Square and A. Pentagon Are Products 160
4. Some Products where One Factor Is a Circle 162
5. Examples of Spaces Incorrectly Expressed as Products 164
6. The Topological Uniqueness of Products 166
7. The Dimension of Product Spaces 167
8. Visualizing S2 x S1 and nT2 x S1 168
9. Geometric Products 170
10. Geometric Products of Flat Spaces 171
11. A Flatland-Friendly Geometric S1 x I 173
12. Flat 3-Dimensional Spaces as Geometric Products 174
13. A Geometric nT2 x S1 177
14. A Geometric S2 x S1 179
15. Isotropic and Non-isotropic Spaces 180
16. Exercises 181
Chapter 8. Geometries of Surfaces 185
1. Euclid’s Axioms 185
2. Flat Surfaces and Euclidean Geometry 191
3. Some Alternative Axioms 192
4. Spherical Trigonometry 193
5. The Area of a Disk in a Sphere 196
6. Maps of the Earth 199
7. Hyperbolic Geometry 200
8. A. Square Learns to Draw a Hyperbolic Plane 202
9. Homogeneous Geometries for all nT2 with n 2 205
10. A Homogeneous Geometry for P2 207
11. Uniqueness of Homogeneous Geometries for Surfaces 208
12. Exercises 209
Part 2. Knots
Chapter 9. Introduction to Knot Theory 215
1. 1-Dimensional Universes 215
2. When Are Two Knots Equivalent? 217
3. The Mirror Image of a Knot or Link 219
4. The Connected Sum of Two Knots 220
5. A Brief History of Knot Theory 221
6. Reidemeister Moves 223
7. Coloring Knots with Three Colors 226
8. Tricolorability and Knot Equivalence 229
9. Oriented Knots and Invertibility 232
10. Connected Sums of Non-invertible Knots 233
11. Exercises 235
Chapter 10. Invariants of Knots and Links 239
1. What’s an Invariant? 239
2. Crossing Number, Tricolorability, and Number of Components 241
3. Positive and Negative Crossings 242
vin
CONTENTS
4. Writhe 244
5. Linking Number 245
6. Nugatory Crossings and Alternating Knots and Links 247
7. Tait’s Conjectures about Alternating Knots 248
8. What Proportion of Knots Are Alternating? 250
9. Seifert Surfaces 252
10. The Genus of a Knot 254
11. Using Euler Characteristic to Compute Genus 257
12. A Potpourri of Knot Invariants 258
13. Exercises 263
Chapter 11. Knot Polynomials 269
1. An Introduction to Polynomial Invariants 269
2. The Rules for the Bracket Polynomial 270
3. The Bracket Polynomial and Reidemeister II Moves 272
4. The Bracket Polynomial with Only One Variable 274
5. The Bracket Polynomial and Reidemeister I Moves 275
6. The X-Polynomial 277
7. The Jones Polynomial 279
8. The State Model for Computing the Bracket Polynomial 282
9. Exercises 285
Part 3. Molecules
Chapter 12. Mirror Image Symmetry from Different Viewpoints 291
1. Mirror Image Symmetry 291
2. Geometric Symmetry 293
3. Geometric Chirality and Achirality 294
4. Chemical Chirality and Achirality 295
5. Chemical Achirality and Geometric Chirality of Figure 8 298
6. Euclidean Rubber Gloves 301
7. Geometrically Chiral and Achiral Knots 302
8. A Topological Rubber Glove 304
9. Topological Chirality 306
10. Exercises 308
Chapter 13. Techniques to Prove Topological Chirality 311
1. Topological Chirality 311
2. Molecular Knots 312
3. Technique 1: Knot Polynomials 313
4. Technique 2: 2-Fold Branched Covers 315
5. Technique 3: Chiral Subgraphs 318
6. Some Graph Theory 320
7. Technique 4: A Combinatorial Approach 322
8. Exercises 325
Chapter 14. The Topology and Geometry of DNA 329
1. Synthetic versus Biological Molecules 329
2. The Biology of DNA 330
3. The Problem of Packing DNA into a Cell 332
CONTENTS ix
4. Supercoiling 333
5. Visualizing DNA as a Circular Ribbon 335
6. The Linking Number of the Backbones 337
7. The Average Writhe of the Axis 339
8. The Twist of a Backbone around the Axis 340
9. The Relationship between Linking, Twisting, and Writhing 343
10. Representing the Supercoiling with a Single Number 345
11. Replication 346
12. Site Specific Recombination 348
13. Knotted and Linked Products of Recombination 349
14. An Introduction to Tangles 350
15. Rational Tangles 352
16. Operations on Tangles 354
17. The Tangle Model of Site Specific Recombination 355
18. Applying the Model to the Enzyme Tn3 Resolvase 357
19. Exercises 358
Chapter 15. The Topology of Proteins 361
1. An Introduction to Proteins 361
2. Trapping Knots in Proteins 362
3. Which Knots Occur in Proteins? 364
4. Examples of Knotted and Linked Proteins 366
5. Non-planar Graphs in Metalloproteins 369
6. The Protein Structure Nitrogenase 373
7. Möbius Ladders in Metalloproteins 376
8. Möbius Ladders in Small Proteins 378
9. Goodbye for Now 380
10. Exercises 381
Index 383
This book is an elementary introduction to geometric topology and its applica-
tions to chemistry molecular biology and cosmology It does not assume any
mathematical or scientific background, sophistication, or even motivation to
study mathematics. It is meant to be fun and engaging while drawing students
in to learn about fundamental topological and geometric ideas. Though the
book can be read and enjoyed by nonmathematicians, college students, or even
eager high school students, it is intended to be used as an undergraduate text-
book.
The book is divided into three parts corresponding to the three areas referred to
in the title. Part 1 develops techniques that enable two- and three-dimensional
creatures to visualize possible shapes for their universe and to use topological
and geometric properties to distinguish one such space from another. Part 2 is
an introduction to knot theory with an emphasis on invariants. Part 3 presents
applications of topology and geometry to molecular symmetries, DNA, and
proteins. Each chapter ends with exercises that allow for better understanding
of the material.
The style of the book is informal and lively. Though all of the definitions and
theorems are explicitly stated, they are given in an intuitive rather than a
rigorous form, with several hundreds of figures illustrating the exposition. This
allows students to develop intuition about topology and geometry without
getting bogged down in technical details. |
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| spelling | Flapan, Erica 1956- Verfasser (DE-588)120957876 aut Knots, molecules, and the universe an introduction to topology Erica Flapan with Maia Averett, Lance Bryant [und 16 weiteren] Providence, Rhode Island American Mathematical Society [2016] xvii, 386 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Topology Textbooks Algebraic topology Textbooks Knot theory Textbooks Geometry Textbooks Molecular biology Textbooks Topologie (DE-588)4060425-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Topologie (DE-588)4060425-1 s DE-604 Averett, Maia Verfasser (DE-588)1168394376 aut Bryant, Lance Verfasser aut Erscheint auch als Online-Ausgabe 978-1-4704-2819-8 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028827235&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028827235&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Flapan, Erica 1956- Averett, Maia Bryant, Lance Knots, molecules, and the universe an introduction to topology Topology Textbooks Algebraic topology Textbooks Knot theory Textbooks Geometry Textbooks Molecular biology Textbooks Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4060425-1 (DE-588)4123623-3 |
| title | Knots, molecules, and the universe an introduction to topology |
| title_auth | Knots, molecules, and the universe an introduction to topology |
| title_exact_search | Knots, molecules, and the universe an introduction to topology |
| title_full | Knots, molecules, and the universe an introduction to topology Erica Flapan with Maia Averett, Lance Bryant [und 16 weiteren] |
| title_fullStr | Knots, molecules, and the universe an introduction to topology Erica Flapan with Maia Averett, Lance Bryant [und 16 weiteren] |
| title_full_unstemmed | Knots, molecules, and the universe an introduction to topology Erica Flapan with Maia Averett, Lance Bryant [und 16 weiteren] |
| title_short | Knots, molecules, and the universe |
| title_sort | knots molecules and the universe an introduction to topology |
| title_sub | an introduction to topology |
| topic | Topology Textbooks Algebraic topology Textbooks Knot theory Textbooks Geometry Textbooks Molecular biology Textbooks Topologie (DE-588)4060425-1 gnd |
| topic_facet | Topology Textbooks Algebraic topology Textbooks Knot theory Textbooks Geometry Textbooks Molecular biology Textbooks Topologie Lehrbuch |
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