#### Volume 6, issue 4 (2006)

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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 ${\pi }_{1}X\cong ℤ∕rℤ$ of order $s\le r$ contains a connected orientable surface $S$ properly embedded in its exterior $X-N\left(K\right)$ 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 $s\ge 4g-1$. 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
Primary: 57M27
Secondary: 57M25