Knot complements, hidden symmetries and reflection orbifolds

Michel Boileau; Steven Boyer; Radu Cebanu; Genevieve S. Walsh

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arXiv · 1501.02253

Knot complements, hidden symmetries and reflection orbifolds

Michel Boileau·Steven Boyer·Radu Cebanu·Genevieve S. Walsh

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Abstract

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.

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Michel Boileau, Steven Boyer, Radu Cebanu, Genevieve S. Walsh. 2015-01-09. Knot complements, hidden symmetries and reflection orbifolds. https://arxiv.org/abs/1501.02253

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