# Can a generic rope exert the same force as a rope with loops?

This is a spin-off of my previous question. I will take a moment to reintroduce the notation. The problem concerns elastic "ropes" in $$\mathbb{R}^3$$, which are modeled as sequences of points $$\gamma=(x_1,x_2,\dots,x_m)$$. A rope is supported on the union of line segments $$S(\gamma) := \bigcup_{j=1}^{m-1} \overline{x_jx_{j+1}} \subset\mathbb{R}^3.$$ The $$j$$-th segment of the rope has direction $$\tau_j := \frac{x_{j+1} - x_j}{|x_{j+1}-x_j|},$$ and the force $$F_\gamma$$ associated to the rope $$\gamma$$ is described by the vector-valued measure $$F_\gamma := \sum_{j=1}^{m-1} \tau_j (\delta_{x_{j+1}} - \delta_{x_j}).$$

This time I am curious about "generic" ropes $$\gamma$$. We say that $$\gamma$$ is generic if the points $$\{x_1,x_2,\dots,x_m\}$$ are in general position. That is, the points are all distinct, no three are collinear, and no four are coplanar. In particular, no two of the line segments $$\overline{x_jx_{j+1}}$$ and $$\overline{x_kx_{k+1}}$$ intersect except when $$j=k+1$$ or vice versa (so there are no loops in $$S_\gamma$$).

Question: Suppose that $$\gamma$$ and $$\gamma'$$ are two ropes with the same force, so that $$F_\gamma = F_{\gamma'}$$. If $$\gamma$$ is generic, does it follow that $$\gamma'$$ is also generic?

What I believe I can show is that, if $$\gamma'$$ is not generic, it must have at least three loops. I would also be curious if there was a reasonable strengthening of the "generic rope" condition that satisfied this conjecture. So what I am really after is something like the question below.

Question' Is there a set $$S$$ of "good ropes" that can be described explicitly with the following properties:

• For two ropes $$\gamma,\gamma'$$ with $$\gamma\in S$$ and $$F_\gamma=F_{\gamma'}$$, $$\gamma'$$ is guaranteed not to have any self-intersections (that is, $$S(\gamma')$$ has no loops).
• The set $$S$$ is "generic" or at least "dense".

For example, I am happy to consider only ropes that do not have any corners with angle $$2\pi/3$$, if that were to help at all.