Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ such that $f|_{\partial D^2}=\gamma$.
It is easy to see that if $\gamma$ admits sealing by the disc then the rotation number of $\gamma$ is $\pm1$. But how can I understand whether or not a given curve admits sealing by the disc?
More surprisingly, there exists curves admitting several inequivalent ways to seal it up by the disc. For example, this one:
Here is the first way:
And here is the second way:
In fact, this curve may be drawn on a higher genus surface instead the plane. Does any theory tell us when we can seal it up by the disc (or, maybe, by the surface of largest genus) and how many ways we can do it? I would like any references about this problem.