Neumann–Reid conjecture on hyperbolic knots with hidden symmetries
A hyperbolic knot is a knot in whose complement admits a complete finite-volume hyperbolic structure. The figure-eight knot and the two dodecahedral knots are the knots of Aitchison and Rubinstein described in the source. A hyperbolic knot admits hidden symmetries if it has an isometry between finite-degree covers of its complement that is not the lift of an isometry of the knot complement.
Neumann–Reid conjecture. The figure-eight knot and the two dodecahedral knots are the only hyperbolic knots in admitting hidden symmetries.
Only these three hyperbolic knots were known to admit hidden symmetries, and they are the only examples known among knots with at most 15 crossings. The conjecture remains open in the supplied source.
References
Primary source
Michel Boileau, Steven Boyer, Radu Cebanu and Genevieve S. Walsh, “Knot complements, hidden symmetries and reflection orbifolds”, arXiv:1501.02253 (2015).
Additional references
2 papers in this index state this conjecture (2010–2015). The statement above is taken from the most recent of them; the others are arXiv:1008.1034.
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 0
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