Non ambient isotopic knots with same Seifert matrix

Question

I am looking for an example of two knots $K,J$ and Seifert surfaces $F,G$ of $K$ respectively $J$, such that for an appropriate basis both surfaces admit the same Seifert matrix, but $K$ and $J$ are not ambient isotopic. I know examples of different Seifert surfaces, that is non ambient isotopic Seifert surfaces, of the SAME knot that admit the same Seifert matrix. However I am looking for an example where $K$ and $J$ are non ambient isotopic. Thanks in advance!
Edit: It suffices to have two non ambient isotopic knots with $S$-equivalent Seifert matrix. I already have proven that then there are appropriate Seifert surfaces of those two knots with the same Seifert matrix.

Answer 1

The existence of such two knots follows immediately from Trotter's Theorem that the Seifert matrix of any knot with trivial Alexander polynomial is $S$-equivalent to the $0\times 0$ Seifert matrix of the unknot together with the existence of non-trivial knots with trivial Alexander polynomial.