# Linking of three triangles in 3-space

## TL;DR

The paper explores the classification and isotopy of triples of disjoint triangles in 3D space, proposing conjectures on algorithmic checks and classification into five types, with an elementary proof distinguishing these types.

## Contribution

It introduces conjectures on algorithmic recognition of isotopy classes and classifies triples of disjoint triangles into five distinct types with a proof distinguishing them.

## Key findings

- Triples of different types are not combinatorially isotopic.
- Conjecture on algorithmic checkability of isotopy.
- Classification of triples into five types.

## Abstract

Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unordered) triple of triangles is combinatorially isotopic to a triple of triangles having pairwise disjoint convex hulls. We also conjecture that any unordered triple of pairwise disjoint triangles in 3-space belongs to one of the 5 types of such triples listed in the paper. We present an elementary proof that triples of different types are not combinatorially isotopic.