## Abstract

The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G = SU(2) these representations (denoted V _{A}(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r^{th} root of unity (r = k + 2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group M(Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of M(Y). (Note that M(Y) has non-trivial center only if Y is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus = 2.) Specifically, for a non-central h ε M(Y) there is an r_{0}(h) such that if r ≥ r_{0}(h) and A is a primitive 4r^{th} root of unity then h acts projectively nontrivially on V_{A}(Y). Jones' [9] original representation ρ_{n} of the braid groups B_{n}, sometimes called the generic q-analog-SU(2)-representation, is not known to be faithful. However, we show that any braid h ≠ id ε B_{n} admits a cabling c = c_{1}, . . . , c_{n} so that ρ_{N}(c(h)) ≠ id, N = c_{1} + . . . + c_{n}.

Original language | English |
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Pages (from-to) | 523-539 |

Number of pages | 17 |

Journal | Geometry and Topology |

Volume | 6 |

DOIs | |

State | Published - 2002 |

Externally published | Yes |

## Keywords

- Jones-Witten theory
- Mapping class groups
- Quantum invariants

## ASJC Scopus subject areas

- Geometry and Topology