## Knot Theory, Volume 24Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject - the Conway, Jones and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology. |

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### Contents

A CENTURY OF KNOT THEORY | 1 |

What Is A Knot? | 11 |

Diagrams and Projections | 24 |

Colorings | 32 |

Matrices Labelings | 42 |

The Alexander Polynomial | 48 |

GEOMETRIC TECHNIQUES | 55 |

Surfaces and Homeomorphisms | 62 |

The Fundamental Group | 105 |

Seifert Matrices and the Alexander | 114 |

Knot Groups and the Alexander | 123 |

NUMERICAL INVARIANTS | 129 |

Braids and Bridges | 135 |

Relations between the Numerical | 141 |

SYMMETRIES OF KNOTS 151 Chapter 8 SYMMETRIES OF KNOTS | 152 |

The Murasugi Conditions | 167 |

Surgery on Surfaces | 73 |

ALGEBRAIC TECHNIQUES | 83 |

Knots and Groups | 89 |

Equations in Groups and the Group | 99 |

Applications of the Murasugi | 173 |

HIGHDIMENSIONAL KNOT | 179 |

Threedimensional Crosssections | 186 |

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added Alexander polynomial algebraic algorithm appear applied arcs argument bands bound boundary braid bridge calculation called changes Chapter Check choice circle classes classical closed colorable column combinatorial components compute condition connected sum consists construction Corollary corresponding cross-sections crossing curve defined definition deformed denoted described determinant discussed disk distinct double drawn easily edges elements entries equations equivalent Euler characteristic example Exercise fact Find follows function genus give given Hence illustrated in Figure instance integer intersection invariants knot diagram knot theory labeled linking number mathematical methods multiplying nontrivial Note operation oriented original period permutation plane points possible presented prime problem projection proof proved quotient rank Reidemeister moves relation result reversible Seifert matrix Seifert surface sequence Show shown signature simple single slice symmetric Theorem tion trefoil triangle unknot unknotting number variables vertices