Searching for non-order-preserving braids algorithmically

An [Formula: see text]-strand braid is order-preserving if its action on the free group [Formula: see text] preserves some bi-order of [Formula: see text]. A braid [Formula: see text] is order-preserving if and only if the link [Formula: see text] obtained as the union of the closure of [Formula: see text] and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid [Formula: see text], confirms this property and returns a proof that [Formula: see text] is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids [Formula: see text] are not order-preserving for any integer [Formula: see text].