Is a knot in $\mathbb{R}^3$ that lies in a plane necessarily trivial?

Question

A knot in $\mathbb{R}^3$ that lies in a plane is necessarily trivial? Trivial means it is equivalent(of the same knot type) to a circle in a plane.

Answer 1

By planar knot I'd assume you mean a simple closed curve in the plane. In that case, you can appeal to the Jordan Curve Theorem.

Answer 2

As Rob Arthan points out, the Jordan Curve theorem isn't quite enough. For example, you can find a closed curve on a genus two surface that cuts it into two pieces but is not a trivial curve. Instead you need the Schönflies Theorem, which tells you the curve is the boundary of a region homeomorphic to the disk. Sometimes this is given as the definition of the unknot (a knot that bounds a disk). To see this, remove a small open disk from the disk. Then you have a homeomorphic copy of an annulus $S^1\times[0,1]$. Use this annulus to guide an isotopy between the original curve and the unknot.