0 An example concerning Ohtsuki ’ s invariant and the full SO ( 3 ) quautum invariant ∗

Two lens spaces are given to show that Ohtsuki’s τ for rational homology spheres does not determine Kirby-Melvin’s {τ ′ r , r odd ≥ 3} By using partial Kirby-Melvin’s quautum SO(3) invariants {τ ′ r(M), r odd prime > |H1(M,Z)|} [KM], Ohtsuki [O] defined a topological invariant τ(M) = ∞ ∑ n=0 λn(M)(t − 1) ∈ Q[[t− 1]] for rational homology 3-sphere M . R.Lawrence [La] conjectured that λn(M) ∈ { Z, if |H1(M,Z)| = 1 Z[12 , 1 |H1(M,Z)| ], if |H1(M,Z)| > 1 and if r is an odd prime which does not divide |H1(M,Z)|, then {|H1(M,Z)|}rτ ′ r(M) is the r−adic limit of the series ∞ ∑