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We calculate the RT-invariants of all oriented Seifert manifolds directlyfrom surgery presentations. We work in the general framework of an arbitrarymodular category as in [V. G. Turaev, Quantum invariants of knots and3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and theinvariants are expressed in terms of the S- and T-matrices of the modularcategory. In another direction we derive a rational surgery formula, whichstates how the RT-invariants behave under rational surgery along framed linksin arbitrary closed oriented 3-manifolds with embedded colored ribbon graphs.The surgery formula is used to give another derivation of the RT-invariants ofSeifert manifolds with orientable base.

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