Finite type invariants of cyclic branched covers
A knot defines a family of closed 3-manifolds indexed by integers, namely the cyclic branched coverings of the knot. It is a long standing question to find a formula for the Casson-Walker invariant of these manifolds. In this paper, the authors give a formula for this invariant in terms of the residues of a rational function (which measures the 2-loop part of the Kontsevich integral of the knot) and the signature function of the knot. The formula, obtained in the hypothesis that the cyclic cover is a rational homology sphere, reads \[ \lambda(\Sigma_{(M,K)}^p)=\frac{1}{3}\text{Res}_p^{t_1,t_2,t_3}Q(M,K)(t_1,t_2,t_3) +\frac{1}{8}\sigma_p(M,K). \] This formula is obtained as a consequence of another formula that computes the LMO invariant of the cyclic branched covers in terms of a rational invariant of the knot and its signature function.
rational lift of the Kontsevich integral
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