  It is well-known that there is a faithful representation of braid groups on
automorphism groups of free groups, and it is also well-known that free groups
are bi-orderable. We investigate which n-strand braids give rise to
automorphisms which preserve some bi-ordering of the free group rank n. As a
consequence of our work we find that of the two minimal volume hyperbolic
2-cusped orientable 3-manifolds, one has bi-orderable fundamental group whereas
the other does not. We prove a similar result for the 1-cusped case, and have
further results for more cusps. In addition, we study pseudo-Anosov braids and
find that typically those with minimal dilatation are not order-preserving.
