Topology Seminar: Eiko Kin

Braids represent mapping classes of the punctured disk, and hence braids induce automorphisms of the fundamental group of the punctured disk, i.e, automorphisms of the free groups. It is known that the free groups are bi-orderable. We consider which braid preserves some bi-ordering of the free group. Once we know a given braid preserves some biordering of the free group, the fundamental group of the mapping torus by the braid monodromy is bi-orderable. By using a criterion by Perron-Rolfsen together with a technique on the disk twists, we give new examples of links in the 3-sphere whose fundamental groups of the link exteriors are bi-orderable, for example, the Whitehead link, the minimally twisted 4- and 5- chain links. We also give an infinite sequence of pseudo-Anosov braids which do not preserve any bi-orderings of the free groups. As a corollary, it follows that the fundamental group of the Whitehead sister link (i.e, (-2,3,8)-pretzel link) exterior is not bi-orderable.