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 ${\hat{I}}^2$ is order-preserving if and only if the link $L$ obtained as the union of the closure of ${\hat{I}}^2$ and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid ${\hat{I}}^2$, confirms this property and returns a proof that ${\hat{I}}^2$ is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids $\ddot{I}{ƒ}_1\ddot{I}{ƒ}_2^{2m+1}$ are not order-preserving for any integer $m$.

13 pages, 3 figures. V2: small typos fixed