The SL(2,C) Casson invariant for Dehn surgeries on two-bridge knots

TL;DR

This paper computes the SL(2,C) Casson invariant for 3-manifolds from Dehn surgeries on two-bridge knots, using seminorms and surgery formulas, and explores its relation to A-polynomials.

Contribution

It provides explicit computations of the SL(2,C) Casson invariant for surgeries on two-bridge knots and relates these to A-polynomials and their multiplicities.

Findings

The invariant is nontrivial for most nontrivial surgeries on hyperbolic two-bridge knots.

Explicit formulas are derived using Culler--Shalen seminorms and Curtis's surgery formula.

Connections between the invariant, degrees of A-polynomials, and higher multiplicity factors are established.

Abstract

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 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 A-polynomials and use this information to identify factors of higher multiplicity in the $\hat{A}$-polynomial, which is the A-polynomial with multiplicities as defined by Boyer-Zhang.