Are two Seifert surfaces with the same Seifert matrix ambient isotopic?

Question

Are two Seifert surfaces with the same Seifert matrix ambient isotopic? I assume not, but it would be really helpful to have a counter example. Thanks in advance!

Answer 1

Yes, this is too good to be true! Here are two Seifert surfaces for the unknot. They're both from taking a tubular neighborhood of a knot (the first, an unknot, the second, a figure-eight) then cutting out a disk.

They're certainly not ambient isotopic since the $\beta$ curve in the second one is a figure eight, which is not a torus knot. Furthermore, with respect to the $\alpha,\beta$ basis for each, the Seifert matrices for both are $$ \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix} $$ At least, I think I got that right, using the $A_{ij}=\operatorname{lk}(x_i^{-},x_j)$ convention.