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2001 Geodesic Knots in the Figure-Eight Knot Complement
Sally M. Miller
Experiment. Math. 10(3): 419-436 (2001).


We address the problem of topologically characterising simple closed geodesics in the figure-eight knot complement. We develop ways of finding these geodesics up to isotopy in the manifold, and notice that many seem to have the lowest-volume complement amongst all curves in their homotopy class. However, we find that this is not a property of geodesics that holds in general. The question arises whether under additional conditions a geodesic knot has least-volume complement over all curves in its free homotopy class.

We also investigate the family of curves arising as closed orbits in the suspension flow on the figure-eight knot complement, many but not all of which are geodesic. We are led to conclude that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.


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Sally M. Miller. "Geodesic Knots in the Figure-Eight Knot Complement." Experiment. Math. 10 (3) 419 - 436, 2001.


Published: 2001
First available in Project Euclid: 25 November 2003

zbMATH: 1014.53025
MathSciNet: MR1917428

Rights: Copyright © 2001 A K Peters, Ltd.


Vol.10 • No. 3 • 2001
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