Planar-Busting Curves on the Boundary of a Handlebody

Abstract

Let $H_n$ be an orientable handlebody of genus $n$. It has been proved that for $n$ not less than 2, there exists an annulus-busting curve in $∂H_n$. In the present paper, we prove that for $n$ not less than 2, there exists an essential simple closed curve $C$ in $∂H_n$ which intersects each essential planar surface in $H_n$ non-emptily. Furthermore, we show that for $n$ not less than 3, a pants-busting curve must also be an annulus-busting curve.