A Slope invariant and the A-polynomial of knots.

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.