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 ${\pi }_{1}X\cong \mathbb{Z}∕r\mathbb{Z}$ 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 $1$.