Given a closed 1D curve in 3D space, can we form a complete set of invariants which uniquely define the curve? (We are only interested in the shape of the curve not its parameterisation.)
Two curves are said to be the same if they overlap completely. (No rotations or transformations allowed).
For example, comparing two ellipses, a complete set of invariants might consist of the central points, the length of the axis, and the angle.
But we are interested in arbitrary closed curves not just ellipses.
Other invariants could be the length of the curve and (if the curve happened to be in a plane) the area enclosed by the curve.
So for example given a curve defined parametrically $x^\mu(\sigma)$, is it possible to define a complete set of invariants $I_n[x]$ that uniquely define the curve. (One might say that the coordinates $x^\mu$ are already a complete set but these are not invariant under reparameterisation where things like central point, or length of curve are).
A "complete set" means we could reconstruct the curve from the invariants (up to parameterisation.)
If the curve is rectifiable, we can come up with a way of generating a parameterization that would be exactly the same for each path with the same image.
Take the point on the curve with the "smallest" coordinates in a lexicographical sense, i.e. find the point with the smallest $x$-coordinate. If there are several, find the point with the smallest $y$-coordinates among those. Again, if there are several, find the point with the smallest $z$-coordinate among those.
Now parameterize the curve by its length, starting at the "smallest point". There are still two ways of doing this, because you can trace the curve in two directions. Choose the variant for which the following holds: The point you encounter at one third of the curve length shall be smaller (again in a lexicographical sense) than the point at two thirds of the curve length.
Now you have a uniquely defined parameterization for each rectifiable curve. As the curve is continuous and hence square-integrable, you can compress this parameterization (which theoretically holds an uncountable amount of information) even further into three sequences of real numbers, one sequence per coordinate, e.g. by applying the Fourier transform.
I have no idea how to get a complete set of invariants (other than the set of points of the curve) if the curve is not rectifiable. Example: Koch snowflake