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ISSN 1079-6762

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


Author: Robert W. Ghrist
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 91-97
MSC (1991): Primary 57M25, 58F22; Secondary 58F25, 34C35
MathSciNet review: 1350685
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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 templates.


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Additional Information

Robert W. Ghrist
Affiliation: Center for Applied Mathematics, Cornell University, Ithaca NY, 14853
Address at time of publication: Program in Applied and Computational Mathematics, Princeton University, Princeton NJ, 08544-1000; Institute for Advanced Study, Princeton NJ, 08540
Email: rwghrist@math.princeton.edu;   robg@math.ias.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-95-02006-3
PII: S 1079-6762(95)02006-3
Keywords: Knots, links, branched 2-manifolds, flows.
Received by editor(s): June 16, 1995
Additional Notes: The author was supported in part by an NSF Graduate Research Fellowship.
The author wishes to thank Philip Holmes for his encouragement and support.
Article copyright: © Copyright 1995 American Mathematical Society