Existence of a continuous map of a disk with a given boundary image on a surface to its complement in $\mathbb{R}^3$

Question

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.

Answer 1

You basically answered your own question in the last paragraph. Take an unknotted torus, with meridian $\mu$ and longitude $\lambda$, and write $S^3- \nu(T) = A \cup B$. Then $\mu$ is null homologous in (say) $A$, and represents the generator of $H_1(B)$, and vice versa for $\lambda$. Any other simple closed curve on $T$ has homology class $p\mu + q \lambda$, and as long as both $p$ and $q$ are nonzero, represents a non-trivial homology class in both $A$ and $B$. So it can't bound a mapped-in disk either. You can do a similar construction for any surface, knotted or otherwise.