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.