Summary: The Apolynomial has ones in the corners.
D. Cooper D.D. Long \Lambda
1 Definition of the Apolynomial.
The Apolynomial was introduced in  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  (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  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