Knots and Links

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American Mathematical Soc., 2003 - Mathematics - 439 pages
Rolfsen's beautiful book on knots and links can be read by anyone, from beginner to expert, who wants to learn about knot theory. Beginners find an inviting introduction to the elements of topology, emphasizing the tools needed for understanding knots, the fundamental group and van Kampen's theorem, for example, which are then applied to concrete problems, such as computing knot groups. For experts, Rolfsen explains advanced topics, such as the connections between knot theory and surgery and how they are useful to understanding three-manifolds. Besides providing a guide to understanding knot theory, the book offers 'practical' training. After reading it, you will be able to do many things: compute presentations of knot groups, Alexander polynomials, and other invariants; perform surgery on three-manifolds; and visualize knots and their complements.It is characterized by its hands-on approach and emphasis on a visual, geometric understanding. Rolfsen offers invaluable insight and strikes a perfect balance between giving technical details and offering informal explanations. The illustrations are superb, and a wealth of examples are included. Now back in print by the AMS, the book is still a standard reference in knot theory. It is written in a remarkable style that makes it useful for both beginners and researchers. Particularly noteworthy is the table of knots and links at the end. This volume is an excellent introduction to the topic and is suitable as a textbook for a course in knot theory or 3-manifolds. Other key books of interest on this topic available from the AMS are ""The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes"" and ""The Knot Book"".
 

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Review: Knots and Links

User Review  - Dustin Tran - Goodreads

Math 199 - Geometric Topology; Rob Kirby. Ch. 1-5. Somewhat hard to follow and old, ugly typeset font. The formatting is pretty ugly as well, but the concepts are there regardless. Because of the few textbooks on the subject, this book still remains as one of the standard classics. Read full review

Contents

CHAPTER ONE INTRODUCTION
1
CHAPTER TWO CODIMENSION ONE AND OTHER MATTERS
8
Knots in the torus
17
The mapping class group of the torus
26
F Higher dimensions
33
G Connected sum and handlebodies
39
CHAPTER THREE THE FUNDAMENTAL GROUP
47
Torus knots
53
CHAPTER SEVEN INFINITE CYCLIC COVERINGS AND THE ALEXANDER
160
Surgery again
168
Computing the Alexander invariant from the knot grouP174
175
Nontrivial knots in higher dimensions with group Z
185
Alexander invariants of links
191
Brunnian links in higher dimensions
197
CHAPTER NINE 3MANIFOLDS AND SURGERY ON LINKS
233
A Foliations
287

E Regular projections
63
G Chains
71
Horned spheres
81
K Unsplittable links in 4space
88
Generalized spinning
96
B The unknotting theorem
103
Knots in solid tori and Companionship
110
E Applications of the sphere theorem
117
B Higherdimensional Seifert surfaces
127
E linking 0 OOOQOOOOOOOOQOQOOOOQOOIO
145
B Branched coverings
293
Cyclic coverings of 3 branched over the trefoil
304
CHAPTER ELEVEN A HIGHERéDIMENSIONAL SAMPLER
343
APPENDIX A COVERING SPACES AND SOME ALGEBRA IN A NUTSHELL
358
APPENDIX B DEHNS LEMMA AND THE LOOP THEOREM
374
TABLE OF KNOTS AND LINKS
388
REFERENCES
430
INDEX
438
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