The Alexander polynomial and finite type 3-manifold invariants

Using elementary counting methods, we calculate a universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M, satisfying , in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Received: 27 April 1998 / in final form: 8 August 1999