For a Seifert matrix $V$ of a knot $K$, the Alexander module has presentation matrix $V-tV^T$. The determinant of this matrix is the Alexander polynomial, which is the order of the Alexander module. In particular, the Alexander module is a torsion module, and has a linking form, called the Blanchfield pairing. The S-equivalence class of the Seifert matrix is an invariant of the knot, and uniquely characterizes the Blanchfield pairing. There is a bijective correspondence between S-equivalence classes of Seifert matrices and Blanchfield pairings.
Trotter gave examples of knots with the same Alexander polynomial but non-S-equivalent Seifert matrices. My question is what additional information we need to reconstruct the Blanchfield pairing (Seifert matrix up to S-equivalence) from the Alexander polynomial.