# Unique diagram of a spatial arc and the knotting probability

## TL;DR

This paper introduces a method to approximate the projection of a spatial arc by a unique diagram and combines it with a knotting probability invariant to analyze the likelihood of knotting in spatial arcs.

## Contribution

It defines a unique arc diagram approximation for spatial arcs and introduces a knotting probability invariant applicable to all oriented spatial arcs.

## Key findings

- Unique arc diagram approximation for spatial arcs
- Knotting probability as an invariant under diagram isomorphisms
- Framework for analyzing knotting likelihood in spatial arcs

## Abstract

It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the spatial arc and the projection. In a separated paper, the knotting probability of an arc diagram is defined as an invariant under isomorphic arc diagrams. By combining them, the knotting probability of every oriented spatial arc is defined.