Casson invariant of cyclic coverings via eta-invariant and Dedekind sums (Q1971126)

Let \(\Sigma\) be an oriented (integer) homology 3-sphere, and let \(\lambda(\Sigma)\) denote its Casson invariant, see for example [\textit{S. Akbulut} and \textit{J. D. McCarthy}, Casson's invariant for oriented homology 3-spheres -- an exposition, Math. Notes, Princeton 36 (1990; Zbl 0695.57011)]. For a knot \(K\) embedded in \(\Sigma\), let \(\Sigma(K,n)\) denote the \(n\)-fold cyclic covering of \(\Sigma\) branched along \(K\). Suppose that the pair \((\Sigma, K)\) admits a plumbing representation (or equivalently, \((\Sigma,K)\) is a graph knot in the sense of \textit{D. Eisenbud} and \textit{W. Neumann}'s book [Three-dimensional link theory and invariance of plane curve singularities, Ann. Math. Stud. 110 (1985; Zbl 0628.57002)]). If \(\Sigma(K,n)\) is an integer homology 3-sphere, then the author computes the expression \(\lambda(\Sigma (K,n))-n \cdot \lambda (\Sigma)\) in terms of homological invariants of the covering (namely, an eta-type invariant associated with the isometric structure of \(K)\).