Finiteness conjecture for low-volume hyperbolic knot complements with hidden symmetries

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A hyperbolic knot complement is the complement of a knot in S3S^3 admitting a complete finite-volume hyperbolic structure, and its volume is denoted by vol⁡M\operatorname{vol} M. A knot complement has hidden symmetries when it non-normally covers another hyperbolic 33-manifold.

Finiteness conjecture. For any R>0R>0, at most finitely many hyperbolic knot complements have hidden symmetries and volume less than RR.

The conjecture is motivated by results producing infinite families of non-accidental-parabolic hyperbolic knot complements without hidden symmetries, while hidden symmetries are known to be rare among knot complements. Its resolution is not stated in the supplied source.

References

Primary source

Eric Chesebro, Jason DeBlois and Priyadip Mondal, “Generic hyperbolic knot complements without hidden symmetries”, arXiv:1910.04712 (2019).

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