Let $ \phi $ be an embedding of $S^1$ in $ R^3$ or $S^3$.
It is often mentionned (for instance when discussing framed knots) that one can choose a trivialization of the normal bundle $ \nu \phi(S^1)$, that is an identification of it with $ S^1 \times R^2$.
Why is this possible?
This is true because one can classify all vector bundles of any given dimension over $S^1$. There is two bundles non isomorphic of dimension $k$ over $S^1$, the first is $\theta^k$ the trivial bundle and the second is $\theta^{k-1}\oplus \eta$ where $\eta$ is the mobius band (or the line bundle over $P^1\equiv S^1$). Since $\phi$ is an embedding and $TS^1$ is trivial so if $\nu\phi(S^1)\equiv\theta^{1}\oplus \eta $ then $\theta^3\restriction \phi(S^1)\equiv TS^1\oplus\nu\phi(S^1)\equiv\theta^2\oplus\eta.$ contradition. So $\nu\phi(S^1)\equiv\theta^{2}.$