I am trying to understand the Whitney Embedding Theorem. It says that any $n$-manifold that is closed and smooth can be embedded as a smooth $n$-submanifold of $R^{2n}$.
Take a complicated looking knot in $\Bbb{R}^3$. Clearly it is a $1$-manifold that is smooth and closed. Is it obvious that it can be embedded as a closed $1$-submanifold of $\Bbb{R}^2$? I am trying to imagine it, but cannot seem to find an embedding.
You are mixing up the imbedding into $\mathbb{R}^3$ and its implication on the topology of the complement with the manifold itself. A smooth closed $1$-manifold can always be diffeomorphically mapped onto the unit circle in $\mathbb{R}^2$.
You will not be able to recover it's embedding into three space from this (the knot), but that's not what Whitneys theorem is about.