## The Link Invariants of the Chern-Simons Field Theory: New Developments in Topological Quantum Field TheoryThe aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic.
Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany |

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### Contents

1 | |

5 | |

6 | |

10 | |

14 | |

18 | |

Chapter 3 Framing in field theory | 23 |

32 Framed Wilson line operators | 25 |

138 Connected sums | 153 |

139 Mutations | 156 |

Chapter 14 Unitary groups | 161 |

142 Casimir operator | 164 |

143 Composite states | 168 |

144 Pattern links | 172 |

145 Higher dimensional representations | 176 |

146 Polynomial structure | 179 |

Chapter 4 NonAbelian ChernSimons theory | 31 |

42 Oneloop effective action | 34 |

43 Higher order results | 39 |

Chapter 5 Observables and perturbation theory | 42 |

52 Perturbative computations | 44 |

Chapter 6 Properties of the expectation values | 47 |

62 Discrete symmetries | 49 |

63 Satellite formulae | 50 |

Chapter 7 Ordering fermions and knot observables | 58 |

72 Antiperiodic boundary conditions | 60 |

73 Knot observables | 64 |

Chapter 8 Braid group | 66 |

82 Hecke algebra | 69 |

Chapter 9 Rmatrix and braids | 74 |

92 Lie algebras and monodromy representations | 79 |

93 QuasiHopf algebra | 80 |

Chapter 10 ChernSimons monodromies | 83 |

102 Universality of the link invariants | 87 |

103 The inexistent shift | 90 |

Chapter 11 Defining relations | 92 |

Chapter 12 The extended Jones polynomial | 102 |

122 Hopf link | 108 |

123 Trefoil knot | 110 |

124 Figureeight knot | 112 |

125 Connection with the Jones polynomial | 118 |

126 Bracket connection | 120 |

127 Reconstruction theorems | 122 |

Chapter 13 General properties | 127 |

132 Recovered field theory | 130 |

133 Links in a solid torus | 133 |

134 Satellites | 138 |

135 Skein relation | 141 |

136 Projectors | 145 |

137 Borromean rings | 147 |

147 SU3 examples | 181 |

Chapter 15 Reduced tensor algebra | 187 |

152 Outlook | 190 |

153 Representation ring | 192 |

154 The threesphere | 195 |

155 Reduced tensor algebra | 196 |

156 Roots of unity | 201 |

157 Special cases | 204 |

Chapter 16 Surgery on threemanifolds | 208 |

162 Solid tori | 209 |

163 Dehn surgery | 214 |

164 Links in threemanifolds | 217 |

165 Elementary surgeries | 220 |

166 Physical interpretation | 222 |

167 The fundamental group | 224 |

Chapter 17 Surgery and field theory | 228 |

172 Properties of the Hopf matrix | 231 |

173 Elementary surgery operators | 237 |

174 Surgery operator | 243 |

175 Surgery rules and Kirby moves | 246 |

Chapter 18 Observables in threemanifolds | 249 |

182 The manifold RP3 | 258 |

183 Lens spaces | 262 |

184 The Poincaré manifold | 266 |

185 The manifold T2 S1 | 269 |

Chapter 19 Threemanifold invariant | 272 |

192 Values of the invariant | 278 |

Chapter 20 Abelian surgery invariant | 283 |

202 Abelian surgery rules | 289 |

203 Abelian surgery invariant | 292 |

References | 303 |

311 | |