The stick number of rail arcs

Consider two parallel lines $\ell_1$ and $\ell_2$ in $\mathbb{R}^3$. A rail arc is an embedding of $\mathbb{R}^3$ such that one endpoint on $\ell_1$, the other $\ell_2$, its interior disjoint from $\ell_1\cup\ell_2$. Rail arcs are considered up to isotopies, ambient isotopies with each self-homeomorphism mapping onto themselves. When manifolds maps taken piecewise linear category, these called stick arcs. The number a class minimum sticks, line segments p.l. arc, needed create representative. This paper will calculate numbers classes crossing at most 2 use winding invariant infinitely many classes. Each has canonically associated knot classes, under over companions. also introduces sticks whose or companion class. calculated for all 9. multi-component as well lattice