2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146109For 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.19 pages, 1 figure. To appear in Osaka Journal of Mathematics. Minor changes including a shorter title, added references, and a research updateGeometric TopologyCombinatorics57M25; 05C12text