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.