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Branched two-manifolds supporting all links. (English) Zbl 0869.57007

The author’s summary: “We resolve several conjectures of J. Birman and R. F. Williams concerning the knotting and linking of closed orbits of flows on 3-manifolds. Our methods center on the symbolic dynamics of semiflows on branched 2-manifolds, or templates. By proving the existence of “universal templates”, or embedded branched 2-manifolds supporting all finite links, we conclude that the set of closed orbits of any flow transverse to the fibration of the figure-eight knot complement in \(S^3\) contains representatives of every (tame) knot and link isotopy class”.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
37G99 Local and nonlocal bifurcation theory for dynamical systems
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