Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 21F / iii / Solution

A geometric circle is an unknot. Compactify ${\mathbb{R}}^3$ to $S^3$, and carry the circle to $\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1{|}^2+|z_2{|}^2=1,z_2=0\}$. Its complement has coordinates $z_1\in \Delta$ and $z_2=\sqrt{1-|z_1{|}^2}{e}^{i\theta }$, making it homeomorphic to $\Delta \times S^1$. It deformation retracts to $S^1$. Removing the additional point at infinity does not change the fundamental group: van Kampen for a small ball about that point uses a simply connected punctured ball, homotopy equivalent to $S^2$. ThusA meridian linking the circle once generates it. This uses “circle” in its geometric sense; the complement of an arbitrary knotted embedded copy of $S^1$ need not have this group.