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

Ghrist, Robert W.

Electron. Res. Announc. Am. Math. Soc. 1, No. 2, 91-97 (1995).

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

Keywords:

knotting; linking; closed orbits; flows; 3-manifolds

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