 
Summary: THE APOLYNOMIAL HAS ONES IN THE CORNERS
D. COOPER AND D. D. LONG
1. Definition of the Apolynomial
The Apolynomial was introduced in [3] (see also [5]), and we present an
alternative definition here. Let M be a compact 3manifold with boundary a torus T.
Pick a basis , µ of
"
T, which we shall refer to as the longitude and meridian.
Consider the subset RU
of the affine algebraic variety R l Hom (
"
M, SL
#
) having
the property that () and (µ) are upper triangular. This is an algebraic subset of R,
since one just adds equations stating that the bottomleft entries in certain matrices
are zero. There is a welldefined eigenvalue map
(iµ ): RU
, #
given by taking the topleft entries of () and (µ). Thus the closure of the image
