## Osaka Journal of Mathematics

### On the distance between two Seifert surfaces of a knot

#### Abstract

For a knot $K$ in $\mathbb{S}^{3}$, Kakimizu introduced a simplicial complex whose vertices are all the isotopy classes of minimal genus spanning surfaces for $K$. The first purpose of this paper is to prove the $1$-skeleton of this complex has diameter bounded by a function quadratic in knot genus, whenever $K$ is atoroidal. The second purpose of this paper is to prove the intersection number of two minimal genus spanning surfaces for $K$ is also bounded by a function quadratic in knot genus, whenever $K$ is atoroidal. As one application, we prove the simple connectivity of Kakimizu's complex among all atoroidal genus $1$ knots.

#### Article information

Source
Osaka J. Math., Volume 46, Number 1 (2009), 203-221.

Dates
First available in Project Euclid: 25 February 2009

https://projecteuclid.org/euclid.ojm/1235574044

Mathematical Reviews number (MathSciNet)
MR2531146

Zentralblatt MATH identifier
1177.57006

#### Citation

Sakuma, Makoto; Shackleton, Kenneth J. On the distance between two Seifert surfaces of a knot. Osaka J. Math. 46 (2009), no. 1, 203--221. https://projecteuclid.org/euclid.ojm/1235574044

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