How many different 'knot shadows' can a circle embedded in high dimension have? Suppose I take a map $\gamma : S^1 \to \mathbb{R}^n$, where $n > 3$. Then, I can project $\gamma$ onto a 3-dimensional subspace of $\mathbb{R}^n$, and (for generic choice of subspace), I will end up with a knot in $\mathbb{R}^3$. 
Can we say anything about the number of inequivalent knots that we can get from projecting a single $\gamma$ onto different subspaces? For instance, for $n = 6$, we can have at least two inequivalent 'knot shadows', by the following construction. Take $\gamma_1, \gamma_2 : S^1 \to \mathbb{R}^3$ to be two different knots, and let
$$\vec{\gamma}(t) = (\vec{\gamma_1}(t), \vec{\gamma_2}(t))$$
so that the projection onto the first three coordinates gives the knot $\gamma_1$ and the projection onto the second three coordinates gives the knot $\gamma_2$. I've not been able to find a similar construction (for multiple knot shadows) for $n=4$ or $n = 5$, though I believe that it should exist. Also, I have not been able to find a construction that gives three different knot shadows, though I believe that should exist.
 A: For each positive integer $n$, there exists an embedding $f:S^1\to \mathbb{R}^4$ together with a collection of subspaces $V_{\theta_i}$ of $\mathbb{R}^4$ where $\dim V_{\theta_i}=3$ and such that the images of the projections $(\pi_i\circ f)(S^1)$ are distinct knots in $V_{\theta_i}$. Here $\pi_i$ is the projection from $\mathbb{R}^4$ to $V_{\theta_i}$. 
Let $K$ be a knot with diagram $D$ in $\mathbb{R}^2$. Then we can realize an embedding of $K$ into $\mathbb{R}^3$ by embedding everything other than neighborhoods of crossings of $D$ into the plane, and in a neighborhood of a crossing, embed a crossing bubble (as below) into $\mathbb{R}^3$.

The trick to my construction is to embed $K$ into $\mathbb{R}^2\times\{0\}\times\{0\}$ except in neighborhoods of crossings. In a neighborhood of a crossing, embed the crossing ball in such a way that when it is projected to $\mathbb{R}^3\times\{0\}$, we recover the embedding into $\mathbb{R}^3$ described above, and when we embed it into $\mathbb{R}^2\times\{0\}\times\mathbb{R}$, we recover an embedding of the mirror of $K$, that is, we flip the roles of the top strand and bottom strand in each crossing bubble.
One can rotate the $3$-dimensional subspace $\mathbb{R}^3\times\{0\}$ to the $3$-dimensional subspace $\mathbb{R}^2\times\{0\}\times \mathbb{R}$. At each angle $\theta$ of this rotation, we get a $3$-dimensional subspace $V_{\theta}$ of $\mathbb{R}^4$.
Although this is slightly hand-wavy, we can embed the knot so that the crossing changes all occur at distinct angles $\theta_1, \theta_2,\dots,\theta_n$, so that $V_{0}$ contains $K$, $V_{\theta_1}$ contains $K$ with one crossing changed, $V_{\theta_2}$ contains $K$ with two crossings changed, etc. 
So now to complete the question, it just remains to show that there is a sequence of knot diagrams such that changing an arbitrary sequence of crossings will lead to a new knot each time. I suspect there are many such examples, but the standard diagrams of the $(2,n)$-torus knots should suffice. 
