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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.

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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.

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    $\begingroup$ Nonuniqueness was known earlier, before Banks $\endgroup$
    – Moishe Kohan
    Commented Jun 16, 2023 at 1:54
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    $\begingroup$ Of course. I have no idea what the first reference is though. $\endgroup$
    – Gooch
    Commented Jun 16, 2023 at 23:19
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    $\begingroup$ At least W. R. Alford, "Complements of minimal spanning surfaces are not unique," Ann. of Math. 91 (1970) 419–424. For more references, see Kakimizu's 1992 paper. $\endgroup$
    – Moishe Kohan
    Commented Jun 16, 2023 at 23:40
  • $\begingroup$ Thanks. This is definitely not the area of knot theory that I know best $\endgroup$
    – Gooch
    Commented Jun 18, 2023 at 20:25
  • $\begingroup$ Oh, I see, thanks for your answers. I came to know that for a fibered knot $K$, its minimal genus Seifert surface is unique up to isotopy, any Seifert surface $F$ of $K$ that induces a fiberation of $S^3-N(K)$ over $S^1$, with $F$ a fiber, must be of minimal genus. This is my motivation to ask this. $\endgroup$
    – Tsoshamry
    Commented Jun 22, 2023 at 2:10

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