# Definition of a knotted curve

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?

• Look at the simplest knotted curve, a trefoil knot, and see if you can "fill it in with a disk". May 13, 2021 at 17:50
• Perhaps I don't understand your question "But how can we 'fill it in with a disk' and not have intersections?", because it seems to me that this question can be answered by the simplest possible example: the circle $\{(x,y,0) \in \mathbb R^3 \mid x^2 + y^2 = 1\}$ can be filled in with the disk $\{(x,y,0) \in \mathbb R^3 \mid x^2 + y^2 \le 1\}$ and that disk does not have intersections. May 13, 2021 at 18:06

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.