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Geodesic knots in the figure-eight knot complement. (English) Zbl 1014.53025
The paper addresses the problem of topologically characterizing simple closed geodesics in the figure-eight knot complement. This is an empirical investigation that relies on the computer software SnapPea, Snap and Tube. Section 2 of the paper contains the theoretical background of the problem. It discusses in general conditions for a hyperbolic manifold to contain infinitely many geodesics. It also recalls Sakai’s result that the Fox group of a simple closed geodesic in an orientable hyperbolic three-manifold is a free group, and lists various knots that, according to Dubois, have free Fox group.
Section 3 restricts itself to the figure-eight knot, and recalls Thurston’s description of the hyperbolic structure of its complement. It then proceeds with an empirical study of the geodesics in the knot complement using the above mentioned software. The conclusion is that the geodesics often appear to have the lowest volume among all curves in their free homotopy class, although this is not always the case, as shown by an example. Finally, section 4 studies closed orbits in suspension flows. The conclusion is that despite several examples, not all closed orbits in the suspension flow of the monodromy for the figure-eight knot complement are geodesics.

MSC:
53C22 Geodesics in global differential geometry
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
Software:
Snap; SnapPea
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