  Given an ideal triangulation of a connected 3-manifold with non-empty
boundary consisting of a disjoint union of tori, a point of the deformation
variety is an assignment of complex numbers to the dihedral angles of the
tetrahedra subject to Thurston's gluing equations. From this, one can recover a
representation of the fundamental group of the manifold into the isometries of
3-dimensional hyperbolic space. However, the deformation variety depends
crucially on the triangulation: there may be entire components of the
representation variety which can be obtained from the deformation variety with
one triangulation but not another. We introduce a generalisation of the
deformation variety, which again consists of assignments of complex variables
to certain dihedral angles subject to polynomial equations, but together with
some extra combinatorial data concerning degenerate tetrahedra. This "extended
deformation variety" deals with many situations that the deformation variety
cannot. In particular we show that for any ideal triangulation of a small
orientable 3-manifold with a single torus boundary component, we can recover
all of the irreducible non-dihedral representations from the associated
extended deformation variety. More generally, we give an algorithm to produce a
triangulation of a given orientable 3-manifold with torus boundary components
for which the same result holds. As an application, we show that this extended
deformation variety detects all factors of the PSL(2,C) A-polynomial associated
to the components consisting of the representations it recovers.
