Identifying Half-Twists Using Randomized Algorithm Methods

TL;DR

This paper introduces a randomized algorithm to identify half-twists in the braid group, aiming to improve computational efficiency in solving conjugacy problems related to braid group elements.

Contribution

The paper presents a novel randomized algorithm for detecting conjugated generators in the braid group, along with a conjecture to enhance its convergence speed.

Findings

Algorithm successfully identifies half-twists in the braid group.

Conjecture suggests potential improvements in convergence speed.

Method offers a practical approach to complex conjugacy problems.

Abstract

Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of operations needed is extremely large. Meanwhile, random algorithms used to solve difficult problems such as primality of a number were developed, and the random practical methods have become an important tool. We give a random algorithm, along with a conjecture of how to improve its convergence speed, in order to identify elements in the braid group, which are conjugated to its generators for a given power. These elements of the braid group, the half-twists, are important in themselves, as they are the key players in some geometrical and algebraical methods, the building blocks of quasipositive braids and they construct endless sets of generators for the group.