# Rail knotoids

## TL;DR

This paper introduces rail knotoids, a new concept in knot theory, establishing a correspondence between rail isotopy of arcs in three-dimensional space and planar knotoid diagram equivalence, with connections to handlebody knot theory.

## Contribution

It defines rail knotoid diagrams and proves their equivalence to rail isotopy in 3D space, linking it to the knot theory of genus 2 handlebodies.

## Key findings

- Rail isotopy in 3D corresponds to planar diagram equivalence.
- Introduces rail knotoids as a new knot theory concept.
- Connects rail isotopy to handlebody knot theory.

## 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 $2$.