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. In this paper our interests lies in the logarithmic Gauss map of the \(A\)-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 \(K\) or equivalently the slopes of the sides of the Newton polygon of the \(A\)-polynomial \(A_K\). We also present a method to compute 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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