Rail knotoids

Abstract

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

15 pages, 15 figures