Ultraknots and limit knots

Abstract

We prove that every knot type in $\mathbb{R}^3$ can be parametrised by a smooth function $f:S^1\to\mathbb{R}^3$, $f(t)=(x(t),y(t),z(t))$ such that all derivatives $f^{(n)}(t)=(x^{(n)}(t),y^{(n)}(t),z^{(n)}(t))$, $n\in\mathbb{N}$, parametrise knots and every knot type appears in the corresponding sequence of knots. We also study knot types that arise as limits of such sequences.