  In this paper, we define a new algebro-geometric invariant of 3-manifolds
resulting from the Dehn surgery along a hyperbolic knot complement in S^3. We
establish a Casson type invariant for these 3-manifolds. In the last section,
we explicitly calculate the character variety of the figure-eight knot and
discuss some applications, as well as the computation of our new invariants for
some 3-manifolds resulting from the Dehn surgery along the figure-eight knot.
