Example of boundary parallel surfaces with boundary

Question

"boundary parallel annulus" always appears in books/papers on 3-manifold theory. I know a surface is called boundary parallel if there is an isotopy sending it onto a boundary component. But if a surface has boundary (e.g. an annulus), how can it be isotoped onto a boundary component of some 3-manifold? Does it mean that there is an isotopy rel boundary sending the surface onto its image? Can any one give any examples?

Answer 1

A properly embedded annulus in a 3-manifold $M$ is boundary parallel if it can be isotoped rel boundary so that its image is entirely contained in a boundary component of the the 3-manifold. After the isotopy, the annulus will not comprise the entire boundary but will simply be a subset of a boundary component.

There aren't any particularly interesting examples of boundary parallel annuli, because by definition, they are simply subsets of a boundary component up to isotopy. Typically one is more interested by non-boundary parallel annuli, because their existence says something about the topology of a manifold.