A Slope invariant and the A-polynomial of knots
Abstract
The A-polynomial is a knot invariant related to the space of SL2(C) representations of the knot group. In this paper our interests lies in the logarithmic Gauss map of the A-polynomial. 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 AK. We also present a method to compute sK in terms of Alexander matrices and Fox calculus.