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Flows on \(S^ 3\) supporting all links as orbits. (English) Zbl 0849.57005

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.
Reviewer: R.W.Ghrist

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
37G99 Local and nonlocal bifurcation theory for dynamical systems
37C10 Dynamics induced by flows and semiflows
Full Text: DOI EuDML

References:

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