  The A-polynomial is a knot invariant related to the space of
$SL_2(\mathbb{C})$ representations of the knot group. Inspired by the results
of Boyer and Zhang, and Dunfield and Garoufalidis, we observe that the
logarithmic slope of the $A$-polynomial detects the trivial knot. We develop a
homological point of view on this slope by extending the constructions of
Degtyarev, the second author and Lecuona to the setting of non-abelian
representations. It defines a rational function on the character variety, which
unifies various known invariants such as the change of curves in the
Reidemeister function, the modulus of boundary-parabolic representations, the
boundary slope of some incompressible surfaces embedded in the exterior of the
knot $K$ or equivalently the slopes of the sides of the Newton polygon of the
A-polynomial $A_K$. We also present a method to compute $s_K$ in terms of
Alexander matrices and Fox calculus.
