Unreduced A-polynomial distinguishing torus knots

Determine whether the unreduced A-polynomial A_K(M,L) of a knot K in S^3 distinguishes torus knots from all other knots; equivalently, establish that if A_K(M,L) equals A_{T_{a,b}}(M,L) for some torus knot T_{a,b}, then K must itself be a torus knot.

Background

The paper proves that the enhanced A-polynomial \tilde{A}K(M,L) distinguishes torus knots (Theorem 1.2), but notes a gap for the unreduced A-polynomial A_K(M,L) due to the universal abelian factor L−1. Specifically, equality of A-polynomials could arise either from equality of enhanced A-polynomials or from an extra L−1 factor multiplying \tilde{A}{T_{a,b}}(M,L), and the latter case cannot currently be excluded.

Clarifying whether A_K(M,L) itself distinguishes torus knots would strengthen detection results and align with known relationships between A-polynomials and boundary slopes of incompressible surfaces.