Genus bounds bridge number for high distance knots

TL;DR

This paper establishes bounds on the bridge number of high-distance knots in 3-manifolds based on genus bounds of essential surfaces, linking geometric complexity to topological invariants.

Contribution

It introduces new bounds on the bridge number for knots with high-distance bridge surfaces, connecting surface genus to knot complexity.

Findings

Bridge number is bounded by 5 for knots with non-trivial reducing surgery.

Bridge number is bounded by 6 for knots with non-trivial toroidal surgery.

For null-homologous knots with Seifert genus g, the bridge number is at most 4g + 2.

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.