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Flows on $S^3$ supporting all links as orbits
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## Flows on $S^3$ supporting all links as orbits

### Robert W. Ghrist

**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
$S^3$ 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 $S^3$ over $S^1$ contains representatives of every (tame) knot
and link isotopy class. Our methods involve semiflows on branched 2-
manifolds, or {\em templates}.

*Copyright American Mathematical Society 1995*

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#### Article Info

- ERA Amer. Math. Soc.
**01** (1995), pp. 91-97
- Publisher Identifier: S1079-6762-95-02006-X
- 1991
*Mathematics Subject Classification*. Primary 57M25, 58F22; Secondary 58F25, 34C35.
*Key words and phrases*. Knots, links, branched 2-manifolds, flows.
- Received by the editors June 16, 1995
- Communicated by Krystyna Kuperberg
- Comments (When Available)

**Robert W. Ghrist**

Center for Applied Mathematics, Cornell University, Ithaca NY, 14853

*Current address:* Program in Applied and Computational Mathematics,
Princeton University, Princeton, NJ 08544; Institute for Advanced Study, Princeton, NJ 08540.

*E-mail address:* `rwghrist@math.princeton.edu; robg@math.ias.edu`

The author was supported in part by an NSF Graduate Research Fellowship.

The author wishes to thank Philip Holmes for his encouragement
and support.

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