## 2-Knots and Their GroupsTo attack certain problems in 4-dimensional knot theory the author draws on a variety of techniques, focusing on knots in S^T4, whose fundamental groups contain abelian normal subgroups. Their class contains the most geometrically appealing and best understood examples. Moreover, it is possible to apply work in algebraic methods to these problems. Work in four-dimensional topology is applied in later chapters to the problem of classifying 2-knots. |

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### Contents

Knots and Related Manifolds | 1 |

The Knot Group | 15 |

Localization and Asphericity | 36 |

The Rank 1 Case | 52 |

The Rank 2 Case | 70 |

Ascending Series and the Large Rank Cases | 85 |

The Homotopy Type of MK | 107 |

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### Common terms and phrases

2-knot group 2-knot with group 2-twist 3-sphere 4-dimensional 4-manifold A-module abelian group abelian normal subgroup algebraic aspherical assume automor central Chapter classical knot closed 4-manifold closed fibre closed orientable 4-manifold cohomological dimension commutator subgroup conjugacy class conjugate Corollary covering space cyclic branched cover determined dimensional knot exact sequence exterior fibred 2-knot fibred knot finite index finite order free abelian normal free group fundamental group geometry group G hence Hirsch-Plotkin radical HNN extension homeomorphism homology homotopy equivalent homotopy type induces infinite cyclic infinite cyclic cover irreducible isomorphic isotopic Lemma locally-finite normal subgroup manifold mapping torus Math matrix meridianal automorphism module monodromy Moreover n-knot nilpotent nontrivial centre orientation preserving PDf group phism Poincare duality Proof Let quotient ribbon knot s-cobordism Seifert fibred 3-manifold Seifert hypersurface Sn+2 subgroup of finite subgroup of rank surgery Theorem torsion free abelian torus knot trefoil knot trivial weight element