Self-reproducing incompressible annulus cut
= Self-reproducing incompressible annulus cut
Let $P$ be a once-punctured <torus> and let $c$ be a nonseparating essential curve. The <annulus> $c\times[0,1]$ is an <incompressible surface> in $P\times[0,1]$. Cutting gives a <pair of pants> times an interval. Both products are genus-two <handlebodies>, since a <compact> <connected> <surface> with <boundary> times an interval is a handlebody of <genus> $1-\chi(P)$.