Definition of a knotted curve

Question

I don't understand the definition of a knotted curve. It says,

A simple closed space curve is knotted if we cannot fill it in with a disk.

But how can we "fill it in with a disk" and not have intersections? Why else would we be unable to fill it in?

Answer 1

Here's a precise statement: An unknotted curve is the boundary of an embedded disk in 3-space. A knotted curve is a curve that is not an unknotted curve.

By this definition, we can say a curve can be filled in with a disk if it is an unknotted curve (i.e., there exists an embedded disk whose boundary is the given curve).

The function of the disk is that it records a way to "trivialize" the curve. Imagine that the disk is given polar coordinates $(r,\theta)$. The $r=1$ curve is the boundary, and for $r=\epsilon$ for $\epsilon>0$ very small, the curve is obviously unknotted in an intuitive sense. The family of curves as $r$ changes gives an isotopy from the original curve to the intuitively trivial curve.

If you allow self-intersecting disks (immersed disks rather than embedded disks), then every knot is the boundary of a disk. This corresponds to the fact that every knotted curve can be unknotted by dragging the curve through itself some number of times (see unknotting number). Technical note: there's a straightforward way to make a disk that also has Whitney umbrella singularities, which is a failure of the disk to be an immersion, but I think you can drag them to the boundary and eliminate them. Another technical note: this is different from the notion of immersed genus (which Gabai proved is equal to Seifert genus) because it allows the interior of the disk to intersect its boundary.