Index

The trefoil knot is given by a continuous embedding $u\colon S^1\to\mathbb{R}^3$, and the unknot is given by a different continuous embedding $v\colon S^1\to\mathbb{R}^3$. This page shows a homotopy $h$ between $u$ and $v$. At time $t=1/4$ the curve crosses over itself, so the map $w(z)=h(1/4,z)$ is not injective. This means that $h$ is not an isotopy, as considered in knot theory. However, it is a perfectly good homotopy, as considered in algebraic topology.