The -polynomial is a knot invariant related to the space of representations of the knot group. In this paper our interests lies in the logarithmic Gauss map of the -polynomial. We develop a homological point of view on this function 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 or equivalently the slopes of the sides of the Newton polygon of the -polynomial . We also present a method to compute this invariant in terms of Alexander matrices and Fox calculus.
Cite: Bénard, Léo. Florens, Vincent. Rodau, Adrien (2024). "A Slope Invariant and the A-polynomial of knots." New York J. Math. 30.
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