The Casson-Walker invariant for branched cyclic covers of Ssp 3 branched over a doubled knot

In 1985, A. Casson defined an invariant λ for oriented integral homology 3spheres by using representations from their fundamental group into SU{2) [1]. It was extended to an invariant for rational homology 3-spheres by K. Walker [11]. In 1993, C. Lescop [9] gave a formula to calculate this invariant for rational homology 3-spheres when they are presented by framed links and showed that it naturally extends to an invariant for all 3-manifolds. Let L be a link in S and let Σ£ be its n-fold cyclic branched cover. Define λn(L) = λ(Σ£). Then λn becomes an invariant of links. For doubles of knots, torus knots and iterated torus knots, A. Davidow (see [3], [4]) calculated Casson's integer invariant for n-fold branched covers, when Σ •£• is an integral homology sphere. For any links, D. Mullins [10] have succeeded in calculating Casson-Walker's rational valued invariant for 2-fold branched covers, when Y?L is a rational homology sphere. In this paper, using C. Lescop's formula and the result of D. Mullins, we will calculate the Casson-Walker invariant for branched cyclic covers of S 3 branched over the m-twisted double of a knot. We will show the following theorem and corollary.