Automated Reidemeister Moves: A Numerical Approach to the Unknotting Problem

TL;DR

This paper introduces a numerical method using generalized Reidemeister moves to efficiently untangle knots from their projections, successfully simplifying knots with up to 50 crossings.

Contribution

It proposes four generalized moves based on Reidemeister moves and a computer program that efficiently untangles knots using Gauss Code notation.

Findings

Successfully untangles all tested knots up to 50 crossings

Runs efficiently with minimal computation time

Results align with theoretical predictions about move effectiveness

Abstract

In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We define four generalized moves based on the Reidemeister Moves. These moves have the capability of untangling knots which must be made more complicated before they can be simplified. With these moves, we construct a computer program which reads the two-dimensional projection of a knot in its Gauss Code notation and untangles it to the fewest possible number of crossings. Due to the properties of the Gauss Code notation, the program runs efficiently with minimal computation time, compared to currently existing untangling programs. We have tested it on all possible Gauss Codes of up to 50 crossings, and the program successfully determines a simplest possible projection of each knot. The results are consistent with our predictions about how the moves work in untangling a knot.