  For q a root of unity of order 2r, we give explicit formulas of a family of
3-variable Laurent polynomials J_{i,j,k} with coefficients in Z[q] that encode
the 6j-symbols associated with nilpotent representations of U_qsl_2. For a
given abelian group G, we use them to produce a state sum invariant
tau^r(M,L,h_1,h_2) of a quadruplet (compact 3-manifold M, link L inside M,
homology class h_1\in H_1(M,Z), homology class h_2\in H_2(M,G)) with values in
a ring R related to G. The formulas are established by a "skein" calculus as an
application of the theory of modified dimensions introduced in
[arXiv:0711.4229]. For an oriented 3-manifold M, the invariants are related to
TV(M,L,f\in H^1(M,C^*)) defined in [arXiv:0910.1624] from the category of
nilpotent representations of U_qsl_2. They refine them as TV(M,L,f)= Sum_h
tau^r(M,L,h,f') where f' correspond to f with the isomorphism H_2(M,C^*) ~
H^1(M,C^*).
