  The Ptolemy variety for SL(2,C) is an invariant of a topological ideal
triangulation of a compact 3-manifold M. It is closely related to Thurston's
gluing equation variety. The Ptolemy variety maps naturally to the set of
conjugacy classes of boundary-unipotent SL(2,C)-representations, but (like the
gluing equation variety) it depends on the triangulation, and may miss several
components of representations. In this paper, we define a Ptolemy variety,
which is independent of the choice of triangulation, and detects all
boundary-unipotent irreducible SL(2,C)-representations. We also define variants
of the Ptolemy variety for PSL(2,C)-representations, and representations that
are not necessarily boundary-unipotent. In particular, we obtain an algorithm
to compute all irreducible SL(2,C)-characters as well as the full A-polynomial.
All the varieties are topological invariants of M.
