I am considering the following problem:

given an embedded closed surface in $\mathbb{R}^3$ (unknotted) and a non-trivial simple closed curve on it, does there exist a continuous map of a disk to the complement of a surface such that the boundary of a disk maps to this curve. Can one describe all these curves for a given surface?

Remark: in fact, I am looking for embedding of a disk but, due to a Loop theorem, such continuous map can be approximated by embedding.

For example, it is almost (?) obvious that for an unknotted torus there are only two such curves - parallel and meridian. For a first there is a continuous map of a disk to the "outer" component of the complement and for a second - to the "inner" one.