We investigate the behavior of the
Casson invariant for
–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
Casson invariant
for the
–manifolds
obtained by
–Dehn
surgery on such knots. These results are applied to prove nontriviality of the
Casson invariant for
nearly all
–manifolds
obtained by nontrivial Dehn surgery on a hyperbolic
two-bridge knot. We relate the formulas derived to degrees of
–polynomials
and use this information to identify factors of higher multiplicity in the
–polynomial, which
is the
–polynomial
with multiplicities as defined by Boyer–Zhang.
Keywords
Casson invariant, character variety, two-bridge knot
