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Knot complements, hidden symmetries and reflection orbifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, pp. 1179-1201.

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the 3-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also show that a reflection orbifold covered by the complement of an AP knot is necessarily small. Thus when K is an AP knot, the complement of K covers a reflection orbifold exactly when K is either the figure-eight knot or one of the dodecahedral knots.

Dans cet article, nous étudions la conjecture de Neumann et Reid selon laquelle les seuls nœuds hyperboliques dans la sphère S 3 admettant des symétries cachées sont le nœud figure-huit et les deux nœuds dodécaédriques. Les nœuds dont les compléments revêtent des orbifold de réflexions hyperboliques admettent des symétries cachées et nous vérifions la conjecture de Neumann et Reid pour ces nœuds lorsque l’orbifold de réflexions est petit. Nous montrons aussi qu’un orbifold de réflexions revêtu par le complément d’un nœud “AP” est nécessairement petit. Ainsi, lorsque K est un nœud “AP”, le complément de K revêt un orbifold de réflexions si et seulement si K est le nœud figure-huit ou l’un des deux nœuds dodécaédriques.

Published online:
DOI: 10.5802/afst.1480
Michel Boileau 1; Steven Boyer 2; Radu Cebanu 3; Genevieve S. Walsh 4

1 Institut de Mathématiques de Marseille I2M, UMR 7373, Université d’Aix-Marseille, Technopôle Château-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France.
2 Dépt. de math., UQAM, P. O. Box 8888, Centre-ville, Montréal, Qc, H3C 3P8, Canada.
3 Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA.
4 Dept. of Math., Tufts University, Medford, MA 02155, USA.
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Michel Boileau; Steven Boyer; Radu Cebanu; Genevieve S. Walsh. Knot complements, hidden symmetries and reflection orbifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, pp. 1179-1201. doi : 10.5802/afst.1480. https://afst.centre-mersenne.org/articles/10.5802/afst.1480/

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