I was reading some materials about knots, some procedures inspired me to ask this question. Given a knot $K$ in $S^3$, one can use Seifert's algorithm to obtain a surface in $S^3$ whose boundary is $K$. Surfaces in $S^3$ with boundaries $K$ are not unique, you can isotopy or add a handle to such a surface to obtain another surface with identical boundary. The Seifert genus of $K$, denoted $g(K),$ is then defined to be the minimal genus of surfaces in $S^3$ whose boundaries are $K$.
It is natural to ask whether such minimal genus Seifert surfaces are unique. Specifically, let $F_1,F_2$ be two Seifert surfaces of $K$, such that $g(F_1)=g(F_2)=g(K)$. Can we isotopy $F_1$, rel boundary, to obtain $F_2$? Thanks for any answer or reference.
No, typically there are many non-isotopic minimal genus Seifert surfaces for a given knot $K$. These surfaces can be organized into a geometric object called the Kakimizu complex. For example Banks has provided examples where this complex is locally infinite:
https://arxiv.org/pdf/1010.3831.pdf
In particular, these are examples of knots with infinitely many non-isotopic Seifert surfaces.