## Knots and LinksRolfsen'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 - GoodreadsMath 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

1 | |

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 |

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### Common terms and phrases

2—sphere 3—manifold abelian group Alexander invariant Alexander polynomial algebraic ambient isotopy annulus attached automorphism ball bicollared boundary link bounds branch set branched cover chapter classical knot codimension components conjecture connected sum consider construction COROLLARY corresponding cover of S3 covering space defined definition Dehn's Lemma denote diagram disjoint disk embedding equivalent example EXERCISE fibration fibred knot foliation fundamental group genus handlebody higher dimensions homeomorphism homology spheres infinite cyclic cover integer intersection isomorphic knot group knot in S3 knot or link knot theory knot type knots and links lens space linking number locally flat longitude loop manifold meridian nontrivial picture Poincare presentation matrix PROOF PROPOSITION relations REMARK Seifert matrix Seifert surface Show simple closed curve simply—connected slice knot Sn+2 solid torus subset surgery description surgery instructions tame knot theorem topological tori torus knot trefoil trivial knot tubular neighbourhood twist unknotted Verify zero