An Algorithm for Detecting Intrinsically Knotted Graphs

TL;DR

This paper presents an algorithm implemented in Mathematica that detects intrinsically knotted graphs and aids in finding knotless embeddings, advancing the classification of such graphs.

Contribution

It introduces a new algorithm capable of recognizing some or all intrinsically knotted graphs and assists in identifying knotless embeddings, expanding known classifications.

Findings

Expanded the list of known minor minimal IK graphs

Successfully identified knotless embeddings for previously unclassified graphs

Demonstrated the algorithm's effectiveness in graph classification

Abstract

We describe an algorithm that recognizes some (perhaps all) intrinsically knotted (IK) graphs, and can help find knotless embeddings for graphs that are not IK. The algorithm, implemented as a Mathematica program, has already been used by Goldberg, Mattman, and Naimi [6] to greatly expand the list of known minor minimal IK graphs, and to find knotless embeddings for some graphs that had previously resisted attempts to classify them as IK or non-IK.