I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow \mathbb{R}^2$ ? Especially, is a computable invariant for such immersed closed curves ?

3$\begingroup$ The paper "Extensions of codimension one immersions" by C. Pappas (1996) TAMS covers your question, although I suspect your question was answered earlier than that. The Pappas paper may give the appropriate reference. $\endgroup$ – Ryan Budney Sep 18 '14 at 14:13

3$\begingroup$ See this answer: mathoverflow.net/a/68021/1345 $\endgroup$ – Ian Agol Sep 18 '14 at 23:14
Here is a cool example (due I believe independently to Eliashberg, Milnor, and Blank) related to your question. There is an immersion of $S^1$ into $\mathbb{R}^2$ that extends to two different immersions of $D^2$ into $\mathbb{R}^2$. Best done with a picture but I don't know how to up load one. A reference is page 150 of "Topology of Spaces of SImmersions" by Eliashberg and Mishachev. Glueing these two together gives rise to a map from $S^2$ to $\mathbb{R}^2$ with only fold singularities which is not homotopic through such maps to the standard quish the 2sphere onto the plane, which is the topic of the E and M paper.

$\begingroup$ Thanks Ian! Now can you prove your true powers by making the two different immersions evident ;). $\endgroup$ – Tom Mrowka Sep 23 '14 at 19:40