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For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Müller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.

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The A-polynomial is a knot invariant related to the space of SL2(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 AK . We also present a method to compute sK in terms of Alexander matrices and Fox calculus.