## Braids and Coverings: Selected TopicsThis book is based on a graduate course taught by the author at the University of Maryland. The lecture notes have been revised and augmented by examples. The first two chapters develop the elementary theory of Artin Braid groups, both geometrically and via homotopy theory, and discuss the link between knot theory and the combinatorics of braid groups through Markou's Theorem. The final two chapters give a detailed investigation of polynomial covering maps, which may be viewed as a homomorphism of the fundamental group of the base space into the Artin Braid group on n strings. |

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### Contents

Braids and Configuration Spaces | 1 |

1 Geometric braids | 2 |

2 Configuration spaces of ordered finite pointsets and their fibrations | 9 |

3 The braid group as fundamental group of a configuration space | 14 |

4 Artins presentation of the braid group | 18 |

5 Representation of braids as automorphisms of free groups | 25 |

6 The Dirac string problem | 39 |

Exercises | 46 |

3 Geometric characterizations of polynomial covering maps | 91 |

4 Polynomial covering maps and homomorphisms into braid groups | 97 |

5 Characteristic homomorphisms for finite covering mans | 103 |

6 An algebraic classification of the polynomial covering maps | 108 |

7 Embedding finite covering maps into bundles of manifolds | 114 |

Exercises | 118 |

Algebra and Topology of Weierstrass Polynomials | 121 |

1 Complete solvability of equations defined by simple Weierstrass polynomials | 123 |

Braids and Links | 49 |

1 Constructing links from braids | 50 |

2 Representing link types by closed braids A theorem of Alexander | 55 |

3 Combinatorial equivalence of closed braids Markovs theorem | 59 |

4 The group of a link | 66 |

5 Plane projections and braid representations of links | 73 |

Exercises | 79 |

Polynomial Covering Maps | 81 |

1 Weierstrass polynomials and the finite covering maps associated with them | 82 |

2 The canonical nfold polynomial covering map | 86 |

2 Primitives for extensions of rings of continuous functions | 130 |

3 The characteristic algebra of a finite covering map | 133 |

4 Weierstrass polynomials and characteristic algebras | 139 |

5 Some applications of characteristic algebras | 143 |

Exercises | 151 |

A presentation of the abstract coloured braid group | 153 |

Threading knot diagrams | 171 |

185 | |

189 | |

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### Common terms and phrases

arcs Artin braid group Aut(Fn automorphism axis base point braid word braided links canonical n-fold polynomial characteristic algebras characteristic homomorphism choice of overpasses closed braid Cn(C coefficient map combinatorially equivalent commutative complex numbers configuration space continuous function Corollary corresponding covering space defined deformation of type denote diagram discriminant elementary deformation elements embedding equivalence class fibration fibre Figure finite covering map Fn(C Fn(M free group fundamental group geometric n-braids hence homomorphism homotopy class induced inverse isomorphism isotopic Lemma link group links in E3 loop Markov Markov-equivalent Math n-fold covering map n-fold polynomial covering negative edge nontrivial permutation phisms plane pointwise polynomial covering map positive edge projection proof of Theorem proves Theorem pull-back ring root map short exact sequence simple Weierstrass polynomial subgroup Suppose Theorem 3.1 threading tion topological space trefoil knot Weierstrass polynomial P(x