Small genus knots in lens spaces have small bridge number

Abstract

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K, and assume s >= 4g-1. Then with respect to the genus one Heegaard splitting of X, K has bridge number at most 1.

This is the version published by Algebraic & Geometric Topology on 11 October 2006

References in corpus (3)

• Holomorphic disks and genus bounds
• Lens space surgeries and a conjecture of Goda and Teragaito
• Dehn surgeries on knots which yield lens spaces and genera of knots

Cited by in corpus (10)

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• Characterizing slopes for torus knots
• Finite Dehn surgeries on knots in $S^3$
• Khovanov homology and the cinquefoil
• Small Dehn surgery and SU(2)
• Bounds on alternating surgery slopes
• Obtaining genus 2 Heegaard splittings from Dehn surgery
• Non-simple genus minimizers in lens spaces
• Bridge number, Heegaard genus and non-integral Dehn surgery
• Once-punctured tori and knots in lens spaces