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}} subsetmathbb{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.