The Apolynomial has ones in the corners. D. Cooper D.D. Long \Lambda Summary: The A­polynomial has ones in the corners. D. Cooper D.D. Long \Lambda 1 Definition of the A­polynomial. The A­polynomial was introduced in [3] and we present an alternative equivalent definition here. Let M be a compact 3­manifold with boundary a torus T: Pick a basis –; ¯ of ß 1 T which we will refer to as the longitude and meridian. Consider the subset R U of the affine algebraic variety R = Hom(ß 1 M;SL 2 C) having the property that ae(–) and ae(¯) 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 well­defined eigenvalue map ¸ j (¸ – \Theta ¸ ¯ ) : R U \Gamma! C 2 given by taking the top left entries of ae(–) and ae(¯): Thus the closure of the image ¸(C) of an algebraic component C of R U is an algebraic subset of C 2 : In the case that the image closure is a curve, there is a polynomial, unique up to constant multiples, which defines this curve [6] (1.13). The product over all components of R U having this property of the defining polynomials for these curves is the A­polynomial. It is shown in [3] that the constant multiple may be chosen so that the coefficients are integers. The additional requirement that there is no integer factor of the result means that the A­polynomial is defined up to sign. The main new result in this paper is that the coefficients of the A­polynomial Collections: Mathematics