Can real algebraic knots be recovered from their projections? Consider a knot represented by a real algebraic curve $C$ in $\mathbb{R}^3$. For example, this answer on SE gives a representation of the trefoil knot. Given a projection of $C$ onto the plane (sans crossing information), can one recover the knot up to chirality?
 A: No, this is not possible. For example, the MSE answer of user porst17 that you linked works out a set of algebraic equations for the following parametrization of the trefoil:
$$
x = \sin t + 2 \sin 2t, \quad y=\cos t - 2 \cos 2t, \quad z=-\sin 3t.
$$
I plotted this in SageMath using:
 parametric_plot3d((sin(t)+2*sin(2*t), cos(t)-2*cos(2*t), -sin(3*t)), (t,0,2*pi))

Here are screenshots of the four views of this curve the SageMath notebook JMol viewer, first the positive octant view, then the XY, XZ, and YZ, projections, from left to right.

However, we can carry out exactly the same procedure to get a set of algebraic equations for the following parametrization of the unknot (pay close attention to $z$):
$$
x = \sin t + 2 \sin 2t, \quad y=\cos t - 2 \cos 2t, \quad z=-\sin t,
$$
and this has the same projection to the xy plane as the trefoil above.
Here's the SageMath code to plot this unknot:
 parametric_plot3d((sin(t)+2*sin(2*t), cos(t)-2*cos(2*t), -sin(t)), (t,0,2*pi))

And here are the resulting screenshots, in the same order as above.

To compute a set of implicit equations for this unknot, I followed almost exactly the steps of porst17 in the abovementioned answer, and this culminated in the following SINGULAR code:
ring R = 0,(x,y,z,s,c),dp;
ideal I = 4*s*c-x+s, 4*s^2-y+c-2, -z-s, s^2+c^2-1;
ideal J = eliminate(I,sc);

The unknot is therefore the variety corresponding to the ideal $J=\langle P_1,P_2,P_3,P_4\rangle$ with
$$
\begin{aligned}
P_1&=x^2+y^2+4xz-12z^2+4y+3\\
P_2&=16z^3-4yz-x-9z\\
P_3&=4yz^2-y^2-xz+6z^2-4y-3\\
P_4&=8xz^2-2xy+2yz-3x-3z.
\end{aligned}
$$
I don't know how to check if there's a simpler generating set, or e.g. if this curve might be a complete intersection. If anyone has suggestions I can try to work it out since I'm curious.
A: No.  The top figures show projections of two trefoil knots, each of which can be assigned a different (or same) crossings.  Join the different assignments in the different crossings, as shown at the bottom to get two different knots, not related by a mirror reflection.
After all, one can be the un-knot (loop) and another a true knot, and these cannot be related by a mirror symmetry (of course).

