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We derive a simple closed formula for the SL 2 (C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL 2 (C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combining computations of the Culler-Shalen norms with the surgery formula for the SL 2 (C) Casson invariant.

We investigate the behavior of the SL.2; C/ Casson invariant for 3-manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler-Shalen seminorms, and we illustrate this approach by providing explicit computations for double twist knots. We then apply the surgery formula of Curtis [18] to deduce the SL.2; C/ Casson invariant for the 3-manifolds obtained by .p=q/-Dehn surgery on such knots. These results are applied to prove nontriviality of the SL.2; C/ Casson invariant for nearly all 3-manifolds obtained by nontrivial Dehn surgery on a hyperbolic two-bridge knot. We relate the formulas derived to degrees of Apolynomials and use this information to identify factors of higher multiplicity in the y A-polynomial, which is the A-polynomial with multiplicities as defined by Boyer-Zhang. 57M27; 57M25, 57M05

Abstract. We establish a formula for the SL 2 (C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL 2 (C) Casson invariant vanishes for spliced sums along knots in S 3 .