Bridge number, Heegaard genus and non-integral Dehn surgery

TL;DR

This paper establishes a linear bound relating the bridge number and tunnel number of hyperbolic knots in 3-manifolds with non-integral surgeries, based on their position relative to a Heegaard splitting.

Contribution

It introduces a linear function bounding the intersection and complexity measures of knots in hyperbolic 3-manifolds with specific surgery properties.

Findings

Bridge number is at most w(g) for knots in the specified setting.

Tunnel number is at most w(g) + g - 1.

Provides bounds linking knot complexity to Heegaard genus.

Abstract

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.