On the distance between Seifert surfaces

Details On the distance between Seifert surfaces On the distance between Seifert surfaces

2007-01-17

arxiv.org/abs/math/0701468v2

Mathematics

Geometric Topology

19 pages, 1 figure. To appear in Osaka Journal of Mathematics. Minor changes including a shorter title, added references, and

Scientific paper

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.

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