Let $\cal{A}$ denote the family of all twice-differentiable simple open arcs $A$ in $\mathbb{R}^3$ satisfying the following properties:

(1) at each point of every arc $A\in\cal{A}$ the curvature of $A$ is at most $1$;

(2) no perpendicular projection of $A$ to any plane is an injection.

> **Question 1.** What is the infimum of the length $|A|$ of $A\in\cal{A}$ ?
> 
> **Question 2.** Is there an arc $A_{min}$ of minimum length among all arcs in $\cal{A}$? If so, is it unique (up to isometry)?
>
**Remark.** This question is related to the notion of *rope knots*, see
Hans Stricker (https://mathoverflow.net/users/2672/hans-stricker), Some questions about ideal knots, URL (version: 2017-04-13): https://mathoverflow.net/q/123170