Planar-Busting Curves on the Boundary of a Handlebody (Communications in Mathematical Research, 2021)
Let $H_n$ be an orientable handlebody of genus $n$. It has been proved thatfor $n$ not less than 2, there exists an annulus-busting curve in $∂H_n$. In the presentpaper, we prove that for $n$ not less than 2, there exists an essential simple closedcurve $C$ 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 bean annulus-busting curve.
Keywords: handlebody; planar surface; planar-busting curve; pants-busting curve.