Consider a compact 3-manifold M with boundary consisting of a single torus. The papers [CS1], [CS2] and [CGLS] discuss the variety of characters of SL 2 (C) representations of π 1 (M ), and some of the ways in which the topological structure of M is reflected in the algebraic geometry of the character variety. We will describe in this paper a certain affine algebraic curve D M which is naturally associated to the character variety of M . In the case that M is the complement of a knot K in a homology 3-sphere we may take the basis B to consist of the longitude and meridian of K. With the usual orientation conventions this basis is well-defined modulo the involution which inverts both the longitude and meridian. It will follow from the construction of D M that if the basis B is obtained from B by inverting both generators then the regular maps p B and p B have the same image. Thus A K := A M,B = A M,B is an invariant of the knot.The polynomial A M,B displays, in a striking way, information about the incompressible surfaces in M . This involves the Newton polygon of A M,B , which is the convex hull of the integer lattice points in the plane whose coordinates arise as degrees of monomials in A M,B . Using the main result of [CS1] we show (Theorem 3.4) that "boundary slopes are boundary slopes," that is that the slope of each side of the Newton polygon of A M,B equals the boundary slope of an incompressible surface in M which is associated to an

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space.Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.

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