Minimal Number of Steps in Euclidean Algorithm and its Application to Rational Tangles

TL;DR

This paper analyzes the minimal steps in the Euclidean algorithm and applies this understanding to efficiently untangle rational tangles, showing that certain methods are optimal in step count.

Contribution

It introduces a regular Euclidean algorithm and demonstrates the minimality of the least absolute remainders method, applying this to rational tangle untangling.

Findings

Least absolute remainders method has minimal steps

Regular Euclidean algorithm aligns with subtraction perspective

Optimal untangling strategy for rational tangles

Abstract

We define the regular Euclidean algorithm and the general form which leads to the method of least absolute remainders and also the method of negative remainders. We are going to show that if looked from the perspective of subtraction, the method of least absolute remainders and the regular method have the same number of steps which is in fact the minimal number of steps possible. This enables us to apply the theory in rational tangles to determine the most efficient way to untangle a rational tangle.