On the absolute value of the SO(3)-invariant and other summands of the Turaev-Viro invariant

Abstract

The Turaev-Viro invariant is defined as a certain state sum calculated on an arbitrary simple spine of a 3-manifold. We specify each term of the sum as 0-term, 1-term or 2-term such that each sum of the terms having the same type is an invariant too. The sum of the 0-terms is equal to the square of the modulus of the so-called SO(3)-invariant. In the paper we express the sum of the 0-terms and 2-terms and the sum of the 1-terms via the Turaev-Viro invariants. Tables of values of the invariants are enclosed. The values are presented as polynomials on primitive roots of unity with integer coefficients.