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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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