Fast algorithmic Nielsen-Thurston classification of four-strand braids

TL;DR

This paper presents a quadratic-time algorithm for classifying four-strand braids into Nielsen-Thurston types, leveraging properties of minimal-length reducible braids and reducing curves.

Contribution

It introduces a novel quadratic complexity algorithm for Nielsen-Thurston classification of four-strand braids, based on geometric properties of minimal-length reducible braids.

Findings

Algorithm runs in quadratic time relative to word length.

Validates that minimal-length reducible 4-braids have round or almost round reducing curves.

Provides a new geometric insight into braid reducibility and classification.

Abstract

We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round.