Genus bounds bridge number for high distance knots

Ryan Blair; Marion Campisi; Jesse Johnson; Scott A. Taylor; Maggy Tomova

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arXiv · 1211.4787

Genus bounds bridge number for high distance knots

Ryan Blair·Marion Campisi·Jesse Johnson·Scott A. Taylor·Maggy Tomova

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Abstract

If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an upper bound for the bridge number of K with respect to T. In particular, a nontrivial, aspherical, and atoroidal knot K with such a bridge surface has its bridge number bounded by 5 if K has a non-trivial reducing surgery; 6 if K has a non-trivial toroidal surgery; and 4g + 2 if K is null-homologous and has Seifert genus g.

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BibTeX RIS

Ryan Blair, Marion Campisi, Jesse Johnson, Scott A. Taylor, Maggy Tomova. 2012-11-20. Genus bounds bridge number for high distance knots. https://arxiv.org/abs/1211.4787

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