Jones polynomial of a knot in terms of its Seifert matrix

Question

It is well known that the Alexander polynomial of a knot can be written in terms of the Seifert matrix of the knot by a simple relationship $$\Delta(t)=\det(V^T-tV),$$ where $t$ is a formal variable and $V$ is the Seifert matrix of the given knot.

Question: Is it possible to write the Jones polynomial of a knot in terms of $V$ as in the Alexander polynomial case?

Thanks in advance.

Answer 1

No, it is not possible. There are examples of “S-equivalent knots” $K$ and $L$ with different Jones polynomials. (You can take $K$ to be the unknot, if you like, to simplify the proof.)