On the Absolute Value of the So(3)–invariant and Other Summands of the Turaev–viro Invariant

1. Introduction. It was proved in [S1] and [S2] that each Turaev–Viro invariant T V (M) q for a 3-manifold M is a sum of three invariants T V 0 (M) q , T V 1 (M) q , and T V 2 (M) q (for definition of the Turaev–Viro invariants see [TV]). It follows from the Turaev–Walker theorem (see [T1], [W]) that, up to normalization, T V 0 (M) q coincides with the square of the modulus of the so–called SO(3)-invariant τ e (M) defined in [T2]. For a connection between SO(3)-invariant and the Reshetikhin–Turaev invariants see [KM] and [BHMV]. With a help of suitable normalizations we make the numbers (T V 0 (M) q + T V 2 (M) q) and T V 0 (M) q invariant under removing of 3-balls. That allows us to define these two invariants on a triangulation of a closed 3-manifold M. It is natural to relate the invariants T V N (M), N = 0, 1, 2, to the Turaev–Viro invariants. Here we show that for every 3-manifold M the following holds: