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Casson-Walker invariants of cyclic covers branched along satellite knots

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Tsutsumi, Y. Osaka J. Math. 45 (2008), 149–157 CASSON-WALKER INVARIANTS OF CYCLIC COVERS BRANCHED ALONG SATELLITE KNOTS YUKIHIRO TSUTSUMI (Received February 27, 2006, revised February 26, 2007) Abstract We express the Casson-Walker invariants for the cyclic covering spaces of the three-dimensional sphere branched along satellite knots in terms of companions, patterns, and winding numbers. 1. Introduction Let C be a knot in S3, and K a knot in a tubular neighborhood N (C) of C . Let be a generator of the kernel of i∗ : H1(�N (C); Z) → H1(S3 − ˚N (C); Z), where i is the inclusion. We regard as a simple closed curve on �N (C). Then, there is a unique embedding f : N (C) → S3, up to isotopy, such that the exterior E( f (C)) = S3− f ( ˚N (C)) is the solid torus and f ( ) bounds a disk in E( f (C)). Denote by C0 the core circle of E( f (C)). The knot P = f (K ) is called a pattern knot for K associated to the companion C . The 2-component link P ∪C0 in S3 is called a pattern link, and w = |lk(P , C0)| is called the winding number (cf. Fig. 1). We denote by 6rK the r -fold cyclic covering space over S3 branched along a knot K . In this paper, we present the Casson-Walker invariant of the r -fold cyclic covering space 6rK of S3 branched along a satellite knot K in terms of patterns, companions, and the winding numbers. A Laurent polynomial 3(t) ∈ Z[t , t−1] is called a knot-Alexander polynomial pro- vided that 3(t−1) = 3(t) and 3(1) = 1. Namely, a knot-Alexander polynomial can be written as a finite sum 3(t) = c0 + ∑ i>0 ci (t i + t−i ), where ci ∈ Z and c0 = 1 − 2 ∑ ci . The Alexander polynomial 1K (t) of a knot K in a homology sphere is a knot- Alexander polynomial. Conversely, given a knot Alexander polynomial 3(t) there is a knot K with 1K (t) = 3(t). We also use Conway’s version of the Alexander poly- nomial ∇K (z) = 1 + ∑ i>0 a2i z 2i , where we denote by a2i (K ) the 2i-th coefficient of the Conway polynomial ∇K (z), which is equivalent to 1K (t) via z = t−1=2− t

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