The root extraction problem for generic braids

TL;DR

This paper presents an efficient algorithm for finding the $k$-th root of a braid in the generic case, significantly improving the speed of root extraction for braids with practical applications.

Contribution

It introduces a new algorithm that efficiently computes $k$-th roots of braids in the generic case, with complexity analysis and handling of non-generic cases.

Findings

Generic-case complexity is $O(l(l+n)n^3 ext{log} n)$.

The algorithm reliably finds roots or confirms their non-existence.

Non-generic cases are addressed with an existing algorithm by Sang-Jin Lee.

Abstract

We show that, generically, finding the $k$-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid $x$ on $n$ strands and canonical length $l$, and an integer $k>1$, computes a $k$-th root of $x$, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is $O(l(l+n)n^3\log n)$. The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee.