Parallelogram Newton polygon criterion for torus knots

Ascertain whether a knot K in S^3 is a nontrivial torus knot if and only if the Newton polygon of its unreduced A-polynomial A_K(M,L) is a parallelogram.

Background

For torus knots T_{a,b}, the Newton polygon of A_{T_{a,b}}(M,L) is a parallelogram whose sides have slopes corresponding to the cabling slope ab and the Seifert slope 0, reflecting the factor L−1.

Given the connection between Newton polygon edges and boundary slopes, establishing this characterization would imply a folklore conjecture that nontrivial torus knots are the only knots in S^3 with exactly two boundary slopes.