On the distance between Seifert surfaces

الأصل · EN

For a knot K, 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.