Reshetikhin–Turaev invariants of Seifert 3–manifolds and a rational surgery formula

Abstract

We calculate the RT–invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expressed in terms of the $S$– and $T$–matrices of the modular category. In another direction we derive a rational surgery formula, which states how the RT–invariants behave under rational surgery along framed links in arbitrary closed oriented 3–manifolds with embedded colored ribbon graphs. The surgery formula is used to give another derivation of the RT–invariants of Seifert manifolds with orientable base.