Searching for non-order-preserving braids algorithmically

TL;DR

This paper introduces an algorithm to determine whether a braid is non-order-preserving by analyzing its link complement, and uses it to prove a family of 3-braids are not order-preserving.

Contribution

The paper presents a novel algorithm for confirming non-order-preservation of braids and applies it to an infinite family of 3-braids, providing new insights.

Findings

Algorithm confirms non-order-preservation of given braids.

Proves that the family of braids σ₁σ₂^{2m+1} are not order-preserving.

Demonstrates the effectiveness of the algorithm in braid theory.

Abstract

An $n$-strand braid is order-preserving if its action on the free group $F_n$ preserves some bi-order of $F_n$. A braid $\beta$ is order-preserving if and only if the link $L$ obtained as the union of the closure of $\beta$ and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid $\beta$, confirms this property and returns a proof that $\beta$ is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids $\sigma_1\sigma_2^{2m+1}$ are not order-preserving for any integer $m$.