The A ‐Polynomial has Ones in the Corners

1. Definition of the A‐polynomial 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 1 T , which we shall refer to as the longitude and meridian. Consider the subset R U of the affine algebraic variety R = Hom (π 1 M , SL 2 C) 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 ξ ≡ ( ξ λ × ξ μ ) :   R U → ℂ 2given by taking the top‐left entries of ρ(λ) and ρ(μ). 1991 Mathematics Subject Classification 57M25, 57M50.