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Small genus knots in lens spaces have small bridge number

Kenneth L Baker

Algebraic & Geometric Topology 6 (2006) 1519–1621

arXiv: math.GT/0612427

Abstract

In a lens space X of order r a knot K representing an element of the fundamental group π1Xr of order s r contains a connected orientable surface S properly embedded in its exterior X N(K) such that 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.

Keywords
(1,1)–knots, Berge knots, bridge position, lens space, Scharlemann cycle, thin position
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25
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Publication
Received: 12 June 2005
Accepted: 16 August 2006
Published: 11 October 2006
Authors
Kenneth L Baker
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160, USA