The A-polynomial Has Ones in the Corners

The A-polynomial was introduced in [3] (see also [5]), and we present an alternative definition here. Let M be a compact 3-manifold with boundary a torus T. Pick a basis λ,μ of π " T, which we shall refer to as the longitude and meridian. Consider the subset R U of the affine algebraic variety R ̄Hom(π " M, SL # #) having the property that ρ(λ) and ρ(μ) are upper triangular. This is an algebraic subset of R, since one just adds equations stating that the bottom-left entries in certain matrices are zero. There is a well-defined eigenvalue map