Wild knot

Counterexample

A knot is a topological embedding $K \colon S^1 \to \R^3$. It is tame if there is a homeomorphism $h \colon \R^3 \to \R^3$ such that $h(K(S^1))$ is a closed polygon with finitely many sides; otherwise it is wild.

Let $p$ be a point of $\R^3$. Take a sequence of small trefoil knots, of sizes tending to $0$, placed one after the other along an arc that ends at $p$. Tie them all into one closed curve that passes through $p$, so that every neighbourhood of $p$ contains infinitely many of the small trefoils. The result is a knot, and it is wild. Every point of a polygonal knot has a small ball around it that meets the knot in one unknotted arc. A homeomorphism of $\R^3$ carries this local picture along. At $p$ the knot has no such ball, because every small ball around $p$ contains whole knotted trefoils of the sequence. One shows that this local knotting cannot be undone.