Boundary of a 2-manifold is a closed curve (or a set of closed curves), so I was thinking of reversing this process.
In 2D space, a simple closed curve in a plane can be thought of as a boundary of a region homeomorphic to $B^2$. In 3D space, a simple closed curve may or may not be knotted, and it seems like for any unknot in 3D, there is a 2-manifold so that the unknot is the boundary of the 2-manifold, which is quite similar to the 2D case, although it may not be unique. However, when it comes to a knot, I cannot find such a 2-manifold that has the knot as its boundary.
Is my observation right?
If so, how can I prove it? If not, what are possible counter-examples?
Can we generalize this to higher dimensions?
Seifert gave a constructive proof of a theorem (originally due to Frankl and Pontrjagin) that says for any knot (or more generally, link), you can always find a compact, connected, oriented surface (now called a Seifert surface) for which the given knot (link) is the boundary.
I haven't tried it myself but the program SeifertView will produce visualizations of these surfaces; see this page of examples.