Small genus knots in lens spaces have small bridge number

In a lens space $X$ of order $r$ a knot $K$ representing an element of the fundamental group $π_{1} X ≅ ℤ ∕ r ℤ$ of order $s \leq r$ contains a connected orientable surface $S$ properly embedded in its exterior $X - N (K)$ such that $\partial S$ 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 $s \geq 4 g - 1$ . Then with respect to the genus one Heegaard splitting of $X$ , $K$ has bridge number at most $1$ .

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Keywords

(1,1)–knots, Berge knots, bridge position, lens space, Scharlemann cycle, thin position

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57M25

References

Forward citations

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

Volume 6, issue 4 (2006)

Kenneth L Baker

Abstract