Algorithmic recognition of quasipositive 4-braids of algebraic length three

TL;DR

This paper presents a polynomial-time algorithm, based on Garside theory, to determine if a 4-braid is a product of three conjugate standard generators, extending to more strings for two conjugate powers.

Contribution

It introduces a new polynomial-time algorithm for recognizing quasipositive 4-braids of algebraic length three, improving computational methods in braid group theory.

Findings

Algorithm decides quasipositivity in polynomial time for 4-braids.

Extends to recognizing products of conjugates of powers of generators for any number of strings.

Provides a practical computational tool based on Garside theory.

Abstract

We give an algorithm to decide whether a given braid with four strings is a product of three factors which are conjugates of standard generators of the braid group. The algorithm is of polynomial time. It is based on the Garside theory. We give also a polynomial algorithm to decide if a given braid with any number of strings is a product of two factors which are conjugates of given powers of the standard generators (in my previous paper this problem was solved without polynomial estimates).