On the distance between Seifert surfaces

Makoto Sakuma; Kenneth J. Shackleton

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On the distance between Seifert surfaces

Makoto Sakuma·Kenneth J. Shackleton

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Abstract

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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BibTeX RIS

Makoto Sakuma, Kenneth J. Shackleton. 2008-01-17. On the distance between Seifert surfaces. https://arxiv.org/abs/math/0701468

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