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The A-polynomial is a knot invariant related to the space of$SL_2(\mathbb{C})$ representations of the knot group. In this paper ourinterests lies in the logarithmic Gauss map of the A-polynomial. We develop ahomological point of view on this slope by extending the constructions ofDegtyarev, the second author and Lecuona to the setting of non-abelianrepresentations. It defines a rational function on the character variety, whichunifies various known invariants such as the change of curves in theReidemeister function, the modulus of boundary-parabolic representations, theboundary slope of some incompressible surfaces embedded in the exterior of theknot $K$ or equivalently the slopes of the sides of the Newton polygon of theA-polynomial $A_K$. We also present a method to compute $s_K$ in terms ofAlexander matrices and Fox calculus.

A Slope invariant and the A-polynomial of knots