Flows on ³ supporting all links as orbits

Abstract

We construct counterexamples to some conjectures of J. Birman and R. F. Williams concerning the knotting and linking of closed orbits of flows on 3-manifolds. By establishing the existence of “universal templates,” we produce examples of flows on S3 containing closed orbits of all knot and link types simultaneously. In particular, the set of closed orbits of any flow transverse to a fibration of the complement of the figure-eight knot in S3 over S1 contains representatives of every (tame) knot and link isotopy class. Our methods involve semiflows on branched 2-manifolds, or templates. In this announcement, we answer some questions raised by Birman and Williams in their original examination of the link of closed orbits in the flow on S induced by the fibration of the complement of a fibred knot or link [4]. In their work, they proposed the following conjecture: Conjecture 0.1 (Birman and Williams, 1983).The figure-eight knot (K8) does not appear as a closed orbit of the flow induced by the fibration of the complement of the figure-eight knot. By “the” flow is meant the flow obtained by integrating the gradient of p, where p : S \K8 → S is the unique fibration over S whose monodromy is the pseudoAnosov representative of its isotopy class, with respect to the Nielsen-Thurston classification [16, 9]. We have resolved this conjecture in the negative and, in so doing, have discovered interesting examples of flows on 3-manifolds and semiflows on branched 2-manifolds. More specifically, in Theorem 3.1, we show that any fibration of the complement of the figure-eight knot in S over S induces a flow on S containing every tame knot and link as closed orbits. We include only those definitions and results which are relevant for this announcement, leaving details to a separate work [11]. 1. Template Theory Periodic orbits of a flow are embedded circles. When the flow is three-dimensional, periodic orbits are knots and the collection of periodic orbits forms a link which often is nontrivial. A valuable tool for examining knotted periodic orbits in three dimensional flows is the template construction of Birman and Williams [3, 4]. Received by the editors June 16, 1995. 1991 Mathematics Subject Classification. Primary 57M25, 58F22; Secondary 58F25, 34C35.