Dear Anton,

Some time ago I was looking at this question (which is important for Chekanov's invariants of legendrian links) and the literature is rather scattered (however, look at the work of Charles Titus in MathSciNet).

I found the following paper, but I haven't really taken a good look at it yet.

"When Does a Curve Bound a Distorted Disk?

Jack E. Graver and Gerald T. Cargo
Consider a closed curve in the plane that does not intersect itself; by the Jordan–Schoenflies theorem, it bounds a distorted disk. Now consider a closed curve that intersects itself, perhaps several times. Is it the boundary of a distorted disk that overlaps itself? If it is, is that distorted disk essentially unique? In this paper, we develop techniques for answering both of these questions for any given closed curve in the plane.

Read More: http://epubs.siam.org/doi/abs/10.1137/090767716"