Show that any simple closed polygon in $\mathbb{R^2}$ belongs to the trivial knot type.

Question

Show that any simple closed polygon in $\mathbb{R^2}$ belongs to the trivial knot type.

Could anyone give me a hint for the solution?

Answer 1

The Jordan-Schoenflies theorem extends the Jordan Curve theorem: Every simple-closed curve in the plane is homeomorphic to the circle. The interior and exterior can be mapped homeomorphically to the respective complements.

https://en.wikipedia.org/wiki/Schoenflies_problem

This clearly does not extend to $\mathbb R^3$ which is why there are non-trivial knots in $\mathbb R^3 $