A Slope invariant and the A-polynomial of knots

TL;DR

This paper introduces a homological approach to the A-polynomial of knots, connecting it with boundary slopes, character varieties, and providing a method to compute related invariants using Alexander matrices.

Contribution

It extends the homological framework to non-abelian representations, unifying various invariants and offering a new computational method for boundary slopes.

Findings

Unified various invariants via a rational function on the character variety.

Extended homological constructions to non-abelian representations.

Provided a computational method using Alexander matrices and Fox calculus.

Abstract

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 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 $A_K$. We also present a method to compute $s_K$ in terms of Alexander matrices and Fox calculus.