The $\mathit{SL}(2,{\mathbb C})$ Casson invariant for Dehn surgeries on two-bridge knots

Abstract

We investigate the behavior of the $\mathit{SL}\left(2,ℂ\right)$ 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 [Topology 40 (2001), 773–787] to deduce the $\mathit{SL}\left(2,ℂ\right)$ Casson invariant for the $3$–manifolds obtained by $\left(p∕q\right)$–Dehn surgery on such knots. These results are applied to prove nontriviality of the $\mathit{SL}\left(2,ℂ\right)$ 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 $A$–polynomials and use this information to identify factors of higher multiplicity in the $Â$–polynomial, which is the $A$–polynomial with multiplicities as defined by Boyer–Zhang.