We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern–Simons functional and prove a localization result: the perturbed critical points either approach a compact subset of the $SL(2, \mathbb{C})$ character variety or else ‘escape to infinity’. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localizing critical points. As an application, we obtain formulas for the Euler characteristic and Poincaré polynomial of the stable locus of the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by 4) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold.

In this paper, we extend the definition of the $SL\left( 2,\,\mathbb{C} \right)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$ -degree of the $\widehat{A}$ -polynomial of $K$ . We prove a product formula for the $\widehat{A}$ -polynomial of the connected sum ${{K}_{1}}\#{{K}_{2}}$ of two knots in ${{S}^{3}}$ and deduce additivity of the $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant under connected sums for a large class of knots in ${{S}^{3}}$ . We also present an example of a nontrivial knot $K$ in ${{S}^{3}}$ with trivial $\widehat{A}$ -polynomial and trivial $SL\left( 2,\,\mathbb{C} \right)$ Casson knot invariant, showing that neither of these invariants detect the unknot.