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.
Citation
Makoto Sakuma. Kenneth J. Shackleton. "On the distance between two Seifert surfaces of a knot." Osaka J. Math. 46 (1) 203 - 221, March 2009.
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