Narrow Results

We work on the notions of rail arcs and rail isotopy in ${ℝ}^3$, and we introduce the notions of rail knotoid diagrams and their equivalence. Our main result is that two rail arcs in ${ℝ}^3$ are rail isotopic if and only if their knotoid diagram projections to the plane of two lines which we call rails, are equivalent. We also make a connection between the rail isotopy in ${ℝ}^3$ and the knot theory of the handlebody of genus $2$.

To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs.

Consider two parallel lines ${\ell }_1$ and ${\ell }_2$ in ${ℝ}^3$. A rail arc is an embedding of an arc in ${ℝ}^3$ such that one endpoint is on ${\ell }_1$, the other is on ${\ell }_2$, and its interior is disjoint from ${\ell }_1\cup {\ell }_2$. Rail arcs are considered up to rail isotopies, ambient isotopies of ${ℝ}^3$ with each self-homeomorphism mapping ${\ell }_1$ and ${\ell }_2$ onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.