Finite type invariants of cyclic branched covers

Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parameterized by the positive integers), namely the cyclic branched coverings of the knot. In this paper, we give a formula for the Casson–Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its signature function.