Searching for non-order-preserving braids algorithmically

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 $β$ is order-preserving if and only if the link $L$ obtained as the union of the closure of $β$ and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid $β$, confirms this property and returns a proof that $β$ is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids $σ_1σ_2^{2m+1}$ are not order-preserving for any integer $m$.