2007/01/17 by Makoto Sakuma, Sakuma, Makoto, Kenneth J. Shackleton +1
Computer Science · Mathematics · #05C12 #57M25 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.math/0701468
openalex publication_date 2007/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.